(CNN) Senate Republicans returned to Capitol Hill on Monday evening after still smarting from the dramatic collapse of their Obamacare repeal bill last week.

Many were in agreement: It's simply time to move on.

"If the question is should we stay on health care until we get it done, I think it's time to move on to something else," Sen. Roy Blunt of Missouri said as he walked into a leadership meeting. "Come back to health care when we have more time to get beyond the moment we are in. See if we can put some wins on the board."

Sen. John Thune, the third-ranking GOP leader, said that "until someone shows us how to get that elusive 50th vote, I think it's over."

Finance Committee Chairman Orrin Hatch of Utah, whose panel oversees both health care and tax policy, said he was ready to tackle taxes.

"I think we are moving on to tax reform but that doesn't mean we can't do more than one thing at a time," he said, adding that Senate Republicans should not vote on health care again unless there is "a reasonable chance of success."

The glum outlook from key GOP senators on Monday came just days after Senate Republicans were unable to pass a bill to repeal key provisions of Obamacare. In a surprise move, Republican Sen. John McCain joined Sens. Lisa Murkowski and Susan Collins in voting "no" early Friday, denying Senate Majority Leader Mitch McConnell and President Donald Trump a significant political victory.

Over the weekend, 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."

But those remarks drew swift pushback from Senate Majority Whip John Cornyn.

"I don't think he's got much experience in the Senate as I recall," Cornyn said of Mulvaney. "He's got a big job. He ought to do that job, and let us do our job."

For the Senate, 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 his or her vote.

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?

Two GOP senators -- Bill Cassidy of Louisiana and Rand Paul of Kentucky -- went to the White House on Monday to try to continue negotiations. Cassidy was asked later if he would like to see additional votes on the issue before the Senate recesses in the next few days. He said that decision was entirely up to McConnell, and that getting one of the three "no" senators to change his or her mind would be a tough task.

Meanwhile, fresh calls for bipartisanship came from both sides of the Capitol.

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."

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"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."