By Jake Tapper and Devan Cole, CNN

(CNN) -- Attempted Russian interference in US public affairs has continued as Department of Homeland Security officials and senior leaders have alerted the White House about the risks ahead of the 2020 presidential election -- but it was "like pulling teeth" to get the White House to pay attention, a US government official told CNN Wednesday.

Officials have "spent months and months trying to sound alarm at the White House about the need to take foreign interference more seriously and elevate the issue," the official said, adding that Director of National Intelligence Dan Coats shared those views and feels the administration "was not being forward-leaning enough in notifying Congress and the American people."

Other than one or two principals' meetings before the midterm elections -- and one press conference in which national security adviser John Bolton, then-Homeland Security Secretary Kirstjen Nielsen, FBI Director Christopher Wray and National Security Agency Director Paul Nakasone warned the public of election interference -- "in general, senior White House staff felt it wasn't a good idea to bring up issues related to Russia in front of the President," the government official said.

The government official said DHS and other agencies made repeated requests to set up more Cabinet-level meetings on the subject and Bolton's National Security Council rebuffed the requests.

"They (DHS) kept 'getting the Heisman' from Bolton and company," the official said.

The bigger concern, the official said, was post-2018 midterms and the need to sit down and figure out how to coordinate a better whole-of-government response, "to get everyone out of their silos. But it was like pulling teeth to get the White House to focus the attention needed on this."

Garrett Marquis, a spokesman for the NSC, told CNN that "any suggestion that this administration is giving less than a full-throated effort to secure America's elections is patently false."

CNN has reached to the Office of the DNI for comment.

Raises concerns over future readiness

The lack of urgency by the White House and the desire to keep the issue away from President Donald Trump -- which was first reported by The New York Times -- raises fresh questions about the readiness of US defenses against foreign interference in the next presidential election. On Tuesday, White House senior adviser Jared Kushner downplayed Russian meddling efforts and suggested that US investigations into them were more harmful than the foreign interference.

Last week, special counsel Robert Mueller said in a redacted version of his report on Russian interference in the 2016 election that a Russian troll group engaged in a years-long campaign to sow discord in the US -- and eventually to support Trump's election -- by creating and maintaining fake social media personas and activist organizations designed to look like they were run by real Americans.

Citing interviews with three unnamed senior administration officials and one former senior official, the Times said Nielsen, who left the White House earlier this month, was told by acting White House chief of staff Mick Mulvaney in a meeting this year that Trump "still equated any public discussion of malign Russian election activity with questions about the legitimacy of his victory."

In a statement to CNN regarding the Times' report, Mulvaney said he didn't "recall anything along those lines happening in any meeting" and insisted that the Trump administration has done more to combat Russian meddling than Barack Obama's administration.

"In fact, for the first time in history, state, local, and federal governments have coordinated in all 50 states to share intelligence, we've broadened our efforts to combat meddling by engaging the Department of Homeland Security, the Department of Justice and the FBI among others, and we have even conducted security breach training drills to ensure preparedness," Mulvaney said.

Rep. Ro Khanna, a California Democrat, called the Times' report "concerning" during an interview Wednesday with CNN's John Berman on "New Day." Khanna also said he "absolutely" wants Nielsen to testify before Congress on the issue.

Later in the same program, Republican Sen. Mike Lee told CNN's Alisyn Camerota that Mulvaney's ask "doesn't make a lot of sense," but suggested that the official's statements may have been misunderstood.

"It may well be that what he was saying was, 'Let's find the right time and place and manner in which to bring that up,' and I suspect that that's the case," said Lee, who represents Utah. "If it is the case, that's not terribly troubling."