By Lauren Fox and MJ Lee

CNN

(CNN) -- The future of the Affordable Care Act is more uncertain than ever.

Over the weekend, Republicans were still reeling from their monumental loss after their plans to overhaul Obamacare were dashed in dramatic fashion.

Some, including President Donald Trump, urged the Senate to try again.

"Unless the Republican senators are total quitters, Repeal & Replace is not dead!" Trump tweeted. "Demand another vote before voting on any other bill."

Office of Management and Budget Director Mick Mulvaney said on CNN's "State of the Union" that it is "officially the White House position" that members stay in Washington until they've passed something.

"What you're seeing there is the President simply reflecting the mood of the people," Mulvaney said. "You promised folks you would do this for seven years."

For the Senate, however, valuable time has already been wasted and there is no signal that any member who voted to defeat the bill Friday is going to be ready to change their vote.

Majority Leader Mitch McConnell already spent two months trying to hammer out a health care deal with his conference in endless private meetings and exhaustive lunches. In the end, the only thing he could offer was a "skinny repeal" that did away with Obamacare's mandates and temporarily repealed the medical device tax. McConnell's closing argument was simply to pass something so that Republicans senators could go to conference with the House. That wasn't enough.

Now, McConnell has other things on his plate. The Senate needs to pass a defense bill, raise the debt ceiling and fund the government.

But Trump isn't the only one urging the Senate to keep trying.

On Friday, House Republicans gathered for a conference where they bemoaned the Senate's dysfunction, playing at the top of their gathering Gordon Lightfoot's "The Wreck of Edmund Fitzgerald," a song about a ship disaster.

They played it, one GOP member told CNN, "as an analogy to the Senate quitting on the verge of victory."

Now, Senate Republicans face a daunting choice. Do they go back to the drawing board, give up or reach across the aisle?

Fresh calls for bipartisanship came from both sides of the Capitol.

Sen. Susan Collins of Maine, one of the three Republicans who voted "no" on Friday, said it was time for Congress to search for a "bipartisan solution" to problems like high premiums, unaffordable deductibles and the lack of choices for patients.

Collins called on senators to "produce a series of bills" aimed at addressing pressing issues in health care. She said the first issue the Senate should focus on would be to stabilize the insurance markets.

"I certainly hope the administration does not do anything in the meantime to hasten that collapse," Collins said on "State of the Union."

New Jersey Rep. Leonard Lance, a moderate Republican who had voted against the House bill to repeal Obamacare earlier this year, reacted to the Senate vote by saying it is now time to "work in a bipartisan capacity."

"I would encourage our Democratic colleagues to come to the table," Lance said.

There are signs that Democrats are ready to negotiate. Instead of gloating after the Republican loss, Democratic leaders in the House and Senate had both reached out to House Speaker Paul Ryan by Friday morning. Senate Minority Leader Chuck Schumer had gone as far as silencing collective Democratic cheers on the Senate floor during the 2 a.m. vote when Arizona Republican Sen. John McCain voted down the bill.

Schumer said in a press conference Friday morning that he did that in hopes that Republicans and Democrats "can work together in a bipartisan way."

Even before the loss on Friday morning, there was some bipartisan recognition that it was time for the parties to hold hearings on health care. Sen. Lamar Alexander, the chairman of the Senate's health committee, announced he would hold hearings. And Sen. Ron Johnson, the chairman of the homeland security and government affairs panel, announced Friday he plans to work through his committee as well.

One of the top items on Democrats' agenda is to stabilize the Obamacare exchanges.

It's unclear if the administration is going to be willing to help, however.

Trump has suggested that Republicans could just let the market collapse. Right now, the Trump administration is responsible for making cost-sharing reduction payments to insurers. If the administration stopped, it could lead to more uncertainty.

"He's going to make that decision this week," White House counselor Kellyanne Conway said on "Fox News Sunday." "And that's a decision that only he can make."

On Saturday, Trump tweeted: "If a new HealthCare Bill is not approved quickly, BAILOUTS for Insurance Companies and BAILOUTS for Members of Congress will end very soon!"

Will Senate move?

All eyes are on the Senate. House Republicans, who passed a comprehensive repeal-and-replace bill, say that if something is going to happen on health care it will have to come from across the Capitol.

GOP Rep. Tom Cole said Friday that he did not expect any real push by House Republicans to restart any Obamacare repeal efforts, particularly because the House has such a full calendar in September.

If the party wants to keep health care on the front burner, Cole said, that is entirely on the Senate.

"We're prepared to go forward if the Senate does anything, but it seems to me the first thing that has to happen is the Senate needs to figure out what it can do," Cole told CNN. "I don't think there's appetite in the House to launch anything."