A group of Senate Republicans held a nearly two-hour meeting with President Biden on Monday night to discuss coronavirus relief legislation, emerging to sound upbeat notes about continuing to work together — but with no deal in hand.

Biden has proposed a $1.9 trillion relief package that Democratic leaders want to move quickly through Congress. The 10-member GOP group led by Sen. Susan Collins (R-Maine) countered with a much narrower $618 billion proposal, and appealed to Biden to make good on his campaign promises to seek bipartisan unity.

Following the Oval Office meeting, Collins told reporters outside the White House that Biden had explained his proposal more thoroughly and they shared the same concerns about the impacts of the pandemic.

“It was a very good exchange of views. I wouldn’t say we came together on a package tonight. No one expected that in a two-hour meeting,” Collins said. “But what we did agree to do is follow up and talk further.”

The path ahead is uncertain, given that Democratic leaders in Congress started the process Monday of advancing a budget bill that can unlock special Senate rules allowing Biden’s package to pass with a simple majority vote in the Senate, instead of the 60 votes usually needed — meaning no Republican votes would be necessary.

But for Biden, the meeting with the GOP senators posed a test for a new president who campaigned on his ability to make bipartisan deals — but also faces strong pressure from the left to deliver a big new relief package now that Democrats control both chambers of Congress and the White House.

Sen. Shelley Moore Capito (R-W.Va.) said: Biden “seemed really happy to be in the game of negotiating.”

If he does leave Republicans behind on his first major piece of legislation, that could further harden the partisan divides Biden promised he would try to bridge, and sour chances for bipartisan legislation for the remainder of his first term in office. But negotiating with Republicans could drag out indefinitely with no guarantee of success, even as Democrats are demanding quick action at a precarious moment for the economy and the pandemic.

Although several GOP senators left the meeting with the impression that Biden was open to negotiating some elements of his plan, White House press secretary Jen Psaki said in a statement afterward that Biden had pointed out deficiencies in the GOP proposal and that his proposal, the American Rescue Plan, “was carefully designed to meet the stakes of this moment, and any changes in it cannot leave the nation short of its pressing needs.”

It was Biden’s first in-person meeting at the White House with lawmakers of either party since becoming president.

Just hours before the meeting began, House Speaker Nancy Pelosi (D-Calif.) and Senate Majority Leader Charles E. Schumer (D-N.Y.) announced they were filing budget bills designed to fast-track Biden’s relief package through the Senate, without GOP votes if necessary.

“Congress has a responsibility to quickly deliver immediate comprehensive relief to the American people hurting from covid-19,” they said in a statement. “The cost of inaction is high and growing, and the time for decisive action is now. With this budget resolution, the Democratic Congress is paving the way for the landmark Biden-Harris coronavirus package that will crush the virus and deliver real relief to families and communities.”

Biden also tweeted in support of his proposed relief bill before the meeting began, and called on Congress to pass it “immediately.”

But at the same time, Biden told Republicans that as a senator he voted on upward of 20 bills taken up through the reconciliation process, with some netting bipartisan support — making the case that just because Democrats pursued that path, that didn’t preclude GOP lawmakers from supporting the relief package.

Sen. Bill Cassidy (R-La.) said in an interview that he and others in the GOP group had made the case to Biden against proceeding via the “budget reconciliation” process that skirts the need for GOP votes, arguing this would undercut his calls for unity.

Cassidy said Biden acknowledged this concern about reconciliation, “but without disowning it.”

“There was never a point where he said, ‘Okay, let’s compromise here,’” Cassidy added. “It was ‘Okay, let us have our staffs speak, we will get you the information upon which we are basing our policy decisions, and then why don’t we speak again.’”

With Biden and Vice President Harris in the Oval Office were White House Chief of Staff Ron Klain, presidential counselor Steve Ricchetti and Brian Deese, Biden’s top economic adviser.

In her statement, Psaki said that Biden had “reiterated that while he is hopeful that the Rescue Plan can pass with bipartisan support, a reconciliation package is a path to achieve that end.”

The nonpartisan Congressional Budget Office said Monday that America would not return to its pre-pandemic unemployment rate this decade. However, CBO also projected stronger-than-expected economic growth in 2021, a finding Republicans seized on to argue the economy does not need another $1.9 trillion in government stimulus.

The Senate GOP plan includes a new round of stimulus checks to Americans and an extension of unemployment insurance benefits that are set to expire in mid-March. Both programs would be far more limited under the GOP plan than what Biden has proposed.

The proposal matches Biden’s call for $160 billion in funding for a national vaccination program, increased testing and related health-care spending. It omits any new money for state and local governments — a major Democratic priority included in Biden’s plan — but includes $40 billion for the small-business Paycheck Protection Program.

Republicans have dismissed Biden’s proposal as overly costly in light of $4 trillion already committed by Congress to fighting the pandemic, including $900 billion in December. But the GOP counter-offer came under immediate attack from Democrats, with Schumer criticizing the exclusion of money for state and local governments.

One area where Biden has suggested he is willing to compromise is on the structure of a new round of stimulus checks. The $900 billion relief bill Congress passed in December included $600 stimulus checks to individuals. Biden’s plan includes a new batch of $1,400 checks, bringing the total to $2,000. That would make good on promises Biden made to Georgia voters ahead of a special Senate election in early January that Democrats won, giving them the Senate majority.

Biden’s plan phases out checks for individuals making $75,000 a year and couples making $150,000 a year.

The GOP plan reduces the size of the checks to $1,000 from $1,400, and caps the income eligibility levels at $50,000 for individuals and $100,000 for couples. The GOP plan would also send just $500 to children and adult dependents, compared with $1,400 in the Biden plan.

Because of how dependents are counted in the Biden proposal, some families making $300,000 or more a year could see some benefit from the stimulus checks, leading senators of both parties to suggest the checks should be more targeted.

In a speech earlier Monday to the Maine State Chamber of Commerce, Collins said Biden seemed open to the idea of targeting the payments, but “very wedded to the $1,400.”

“Maybe there’s a compromise right there,” Collins said, “of targeting but going with a bit higher amount.”

Biden’s plan includes an array of other items omitted from the GOP proposal, including rental assistance and eviction forbearance, an increased child tax credit, and an increase in the federal minimum wage to $15 an hour.

Biden’s plan would also increase emergency unemployment benefits from $300 a week to $400 a week and extend them through September; the GOP plan would leave the payments at $300 a week and extend them through June. If Congress does not act, the enhanced unemployment insurance will expire in mid-March.

In addition to Collins, Capito and Cassidy, the senators involved in Monday’s meeting were Sens. Lisa Murkowski (Alaska), Mitt Romney (Utah), Rob Portman (Ohio), Todd C. Young (Ind.), Jerry Moran (Kan.), Thom Tillis (N.C.) and Mike Rounds (S.D.).

If Democrats were to join with the 10 Republican senators, they could reach the 60-vote threshold needed to pass legislation under normal Senate procedures. The Senate is split 50-50 between Republicans and Democrats, with Democrats holding the majority because Harris can break ties.

Even if Democrats moved forward without the support of Republicans through the process known as budget reconciliation, passing Biden’s plan could still be challenging. It would mean they wouldn’t be able to lose a single Democratic vote in the Senate.

Behind the scenes, the office of Sen. Joe Manchin III (D-W.Va.), considered the most conservative Senate Democrat, has been pressing the White House to reduce the size of its $1.9 trillion stimulus proposal, according to two people who spoke on the condition of anonymity to share details of the private exchange. In particular, Manchin’s office has expressed concern about approving more money before the funding approved in the prior relief packages has been spent by the federal government, these people said.

Colby Itkowitz contributed to this report.