By Sara Murray and Kevin Liptak, CNN

(CNN) -- A likely contagious President Donald Trump returned to the White House Monday evening, whipped off his mask and filmed a video, heavy on bluster and short on facts, that proclaimed: "The vaccines are coming momentarily."

Trump, who tested positive for coronavirus last week and is now receiving medical care at the White House, has remained laser-focused on vaccine development even as he has been dismissive of mask-wearing and social distancing -- protections health experts say are critical to stopping the spread of the coronavirus.

Even before his diagnosis, the President had taken to calling drug companies to check on their vaccine trials, asking how much longer they'll take and ginning up the pressure around his desire for a vaccine before Election Day. He's also signaling he might speed up the federal approval process, conspicuously stalling Food and Drug Administration recommendations that would delay a vaccine authorization.

In his conversations with major drug-makers working on coronavirus vaccines, Trump has been explicit in telling the companies' CEOs that he'd like to see a vaccine move quicker than some of his health advisers say is reasonable, according to a person familiar with the conversations. He has asked whether they believe they can speed up their timelines and has suggested he is concerned that the FDA's regulatory process could slow down progress.

Trump's repeated calls -- especially to Pfizer, whose progress the President sounds most hopeful about recently -- are likely to continue through a presidential election that may turn on how voters perceive his handling of the public health crisis.

Meantime, experts worry what Trump's undivided attention means for the fate of the vaccine. With so much obvious political pressure coming to bear, people may fear that the vaccines aren't safe for widespread use. Beyond that, they may lose trust in federal regulators and, possibly, in research science.

The political climate surrounding a potential coronavirus vaccine already has scientists cringing.

"There just seems to be this huge pressure from an administration that has been very effective at getting everything wrong," said Dr. Esther Choo, an emergency medicine physician and professor at Oregon Health & Science University. "So, it's like, how can this go well?"

White House spokeswoman Sarah Matthews dismissed those concerns.

"This President understands that this vaccine cannot get bogged down in government bureaucracy," Matthews said. "The Trump administration is focused on delivering a safe, effective vaccine to the American people as quickly as possible and any efforts to undermine confidence in a vaccine are irresponsible and dangerous."

In the first presidential debate, Trump acknowledged that he sees vaccine development not simply as a scientific matter, but also as a political one.

"It is a very political thing," he said. "I've spoken to Pfizer, I've spoken to all of the people that you have to speak to, Moderna, Johnson & Johnson and others," he said. "They can go faster than that by a lot."

After the late September presidential debate, Pfizer chief executive officer Dr. Albert Bourla penned a letter to employees insisting he would not be swayed by political pressure, adding, "we are approaching our goal and despite not having any political considerations with our pre-announced date, we find ourselves in the crucible of the US presidential election."

Scientists are particularly concerned that Pfizer could play into Trump's electoral timeline. Trump often trumpets the work of the company, musing aloud about their revealing a vaccine before Election Day.

"We think we have a winner there," Trump said of Pfizer's vaccine back in July, as he announced the government would pay the company $1.95 billion to buy the first 100 million doses of its coronavirus vaccine, once it meets government approvals.

The drug company is led by a boastful chief executive who has appeared unafraid to bust the norms around vaccine development.

In various interviews, Bourla has predicted that the company would know by this month whether its vaccine works. And in an interview with The Washington Post in August, Bourla made it clear that his company wouldn't hesitate to seek government approval.

"We would go for regulatory approval in October, and we expect, given how fast the regulatory agencies all over the world are reviewing those applications, that we will do it very quickly," Bourla said.

'I've never seen it'

Experts have been stunned by Bourla's bravado on display.

"To have him independently promising a timeline around vaccines is really -- I've never seen it," said Dr. Tom Inglesby, director of the Center for Health Security of the Johns Hopkins Bloomberg School of Public Health."I think it's particularly unfortunate given the political pressure that's already on the scientific agencies at this point."

Inglesby was one of the dozens of scientists who recently penned a letter to Pfizer asking the company to wait until at least November before it seeks approval for its vaccine.

"Their CEO is the only CEO who's been really outspoken about saying we'll know by October," said Dr. Eric Topol, executive vice president of Scripps Research, who also signed the letter.

"It had all the look of he and Trump were very much in sync," Topol said of Bourla's rosy predictions for an October vaccine. "They could jeopardize the entire vaccine landscape if they do this rush job and get it through."

Amy Rose, a spokeswoman for Pfizer, said the company was trying to be transparent about its timeline but acknowledged that the company is blind to the data being collected during human trials. That data is independently evaluated by a data monitoring committee.

"We are working as fast as possible without cutting corners. We will only submit for authorization when we have data that complies with the FDA's criteria for safety and efficacy," Rose said.

In his conversations with drug company leaders, Trump has appeared keenly interested in receiving updates on their trials and has asked the executives how long they believe it will take to reach the approval stage, according to the person familiar with the discussions.

Trump publicly disclosed on September 4 that he'd spoken to the head of Pfizer in recent days, though the person familiar said he's spoken a number of times over the summer with the heads of Pfizer, Johnson and Johnson and Moderna, all of which have vaccine candidates in various stages of trial.

"I just spoke to the head of Pfizer, great guy. They announced that it expects to have the results of its trial very, very shortly," Trump said in September. He noted a number of vaccines were in the midst of human trials, saying, "I think you're going to see results that are shockingly good."

Rose, the Pfizer spokeswoman, said the company has "regular conversations with the administration on a wide range of health policy matters" including the vaccine but added "it would be inappropriate for us to discuss publicly the content of those private conversations."

Trump doesn't have those reservations. In public, he has sounded most bullish on Pfizer, saying publicly he believes it will be the first to reach some stage of approval.

"There's never been anything like it," Trump said, touting Pfizer's vaccine trial back in July. "It's the fastest ever, and to me, it's very exciting."

In his post-debate letter to employees, Bourla wrote: "We would never succumb to political pressure. The only pressure we feel -- and it weighs heavy -- are the billions of people, millions of businesses and hundreds of government officials that are depending on us."

Unreasonable timeline

Most experts -- including top health officials in the Trump administration -- don't view this month as a reasonable vaccine timeline, instead suggesting that a vaccine may be available to some by late 2020 or early 2021 then widely available midway through next year.

Scientists at the FDA, which is responsible for approving or authorizing vaccines, have insisted a vaccine will be greenlit based on scientific -- not political -- criteria. They have pledged to make the data surrounding vaccine candidates publicly available and to present candidates to an outside review board for evaluation and public discussion.

But the President's unrelenting political pressure has experts fearing he and his political appointees could manipulate the normal approval process.

Those fears were exacerbated recently when the FDA suggested it wanted to tighten its guidance for vaccine companies to require more data, a move that would have likely pushed an authorization beyond Election Day.

"We're looking at that and that has to be approved by the White House. We may or may not approve it," the President said of the new FDA guidelines at the time. "That sounds like a political move."

The guidance is still stalled, but FDA officials have publicly tried to downplay the intervention.

"There's no there there about this guidance to get all excited," Dr. Peter Marks, the FDA's director of the Center for Biologics Evaluation and Research said at a recent nonprofit event for the Friends of Cancer Research. "The companies know what we're expecting."

The President's opposition to more stringent vaccine guidelines doesn't fill scientists with hope that he would leave the broader vaccine process untouched.

Trump administration officials are "both making these decisions that are not correct, in an objective scientific sense, but also are derailing the usual protections against making these decisions that are not scientifically correct," Choo said. "That combination is what's been the perfect storm in this pandemic."

In a typical vaccine authorization process, the vaccine will be evaluated by the FDA's scientists and Marks will send a recommendation to FDA commissioner Dr. Stephen Hahn on whether or not it should be approved.

From there, scientists worry the President could try to sway Hahn's decision. Hahn said last month in congressional testimony that any decision to authorize a vaccine will be made based on science and the recommendations from career FDA officials.

"FDA will not permit any pressure from anyone to change that," Hahn told Congress.

Marks has said he would resign if the President tries to push forward with a vaccine that has not been proven safe and effective. Hahn also has not ruled out the possibility of resigning if he is pressured to authorize a vaccine against the wishes of career FDA scientists.

"I think all options are on the table, with respect. I hope we won't be in that position," Hahn said in an interview with CBS News.

But there's another deeply disturbing possibility floating around among scientists -- one that would seem like little more than outlandish conspiracy theory in a more normal year, under a President less willing to wield his political power over typically apolitical health agencies.

"It would be unprecedented, but in theory the HHS Secretary (Alex Azar) could issue the decision," said Dr. Jesse Goodman, former FDA Food and Drug Administration chief scientist and the director of Georgetown University's Center on Medical Product Access, Safety and Stewardship.

It's perhaps the worst-case scenario as far as health experts are concerned: Azar could use his authority as the head of Health and Human Services to authorize a vaccine without the FDA fully vetting it.

"If they did not follow through with the plans to bring the data and have pubic discussion at FDA's advisory committee, that would be really worrisome," Goodman said.

Brian Harrison, the chief of staff at the Department of Health and Human Services, insisted that wouldn't happen.

"Secretary Azar will defer completely to the FDA as they weigh whether to approve or authorize any particular vaccine or therapeutic," Harrison said in a statement to CNN. "The idea the Secretary would approve or authorize a vaccine over the FDA's objections is preposterous and betrays ignorance of the transparent process that we're following for the development of the OWS vaccines."

Most experts still believe it's extremely unlikely HHS would bigfoot the FDA.

"People call me with all kinds of paranoid theories, but I just don't think that any of them will happen," said Dr. Joshua Sharfstein, a vice dean at the Johns Hopkins Bloomberg School of Public Health and a former principal deputy commissioner at the FDA. "It would be so self-defeating to jump the whole vaccine process, for what? For a vaccine very few people are going to take and no doctors are going to recommend?"

The political jockeying around a coronavirus vaccine may already be taking a toll on Americans' confidence. Some 45% of Americans said they would not try to get vaccinated even if a low-cost vaccine is widely available, according to an October CNN poll conducted by SSRS.

At his recent appearance speaking to a nonprofit group, the FDA's Marks tried to offer reassurances that the health agency would perform its job as it always does to put forth a vaccine as quickly and as safely as possible to save lives.

"The most important thing that I think I can do, and I can help do, in the coming months is to help generate trust -- regain trust in vaccines," Marks said. "Vaccines have saved public health previously. They will save it again. We just are going to have to believe in them."