The Republicans' proposed package is estimated to cost $618 billion, about one-third the size of Biden's $1.9 trillion proposal. It provides $160 billion for vaccines, $132 billion for a smaller unemployment benefit and $1,000 direct payments to Americans, according to an official summary.
Sen. Susan Collins, R-Maine, who is leading the group, called it "a frank and very useful discussion" in brief remarks at the White House.
"It was a very good exchange of views. I wouldn't say we came together on a package tonight. No one expected that," Collins said. "But what we did agree to do is follow up and talk further."
The other members of the group are Sens. Lisa Murkowski of Alaska, Bill Cassidy of Louisiana, Rob Portman of Ohio, Shelley Moore Capito of West Virginia, Todd Young of Indiana, Jerry Moran of Kansas, Mike Rounds of South Dakota, Thom Tillis of North Carolina and Mitt Romney of Utah.
It was not clear if they had made progress toward an agreement.
Also on Monday, House Speaker Nancy Pelosi, D-Calif., and Senate Majority Leader Chuck Schumer, D-N.Y., introduced a budget resolution that would pave the way for Democrats to pass a relief bill without any Republican support, should a deal fail to materialize.
Biden's invitation to the Republican senators came late Sunday after the senators requested the meeting.
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It is a Republican-only proposal at a time when Democrats control the White House and Congress. But it will test Biden's calls for unity and bipartisanship while promising lofty policy goals.
Brian Deese, the director of the National Economic Council, said Sunday on NBC News' "Meet The Press" that Biden "is open to ideas wherever they come."
"What he's uncompromising about is the need to move with speed on a comprehensive plan," he said.
Speaking Sunday on CNN's "State of the Union," Portman said he hopes Biden will meet with the lawmakers and strike a deal rather than have Congress advance legislation through a special budget process known as reconciliation, which would allow Democrats to bypass the Senate's 60-vote threshold and approve a larger relief bill without Republican support.
Portman, who recently spoke with Biden after he announced that he will not seek re-election in 2022, said the amount Republicans would agree to in such a negotiation would be far less than the $1.9 trillion the administration seeks.
"Let's focus on those who are struggling," he said, adding: "We really want to help those who need it the most. And at a time of unprecedented deficits and debts — and a debt, as the percentage of the economy, is as high as it's been in our nation's history since World War II — we need to be sure this is targeted."
Republican lawmakers have largely rejected Biden's $1.9 trillion plan, balking at the price tag. But the new Republican offer is likely to face progressive pushback, as Democrats like Rep. Pramila Jayapal, D-Wash., have described Biden's offer as only a "promising start."
It will be a slog for Biden to secure 10 Republican votes on a major Covid-19 bill. The senators have not yet made public a fully fleshed-out proposal. And just three of the signatories — Collins, Murkowski and Romney — have track records of breaking with their party on politically salient votes.
White House press secretary Jen Psaki, at her Monday news briefing, said the meeting between Biden and the senators would serve as "an opportunity to exchange ideas" but repeatedly indicated that the president wants a larger package.
"His view is that the size of the package needs to be commensurate with the crises we’re facing. That’s why he proposed $1.9 trillion," Psaki said.
A day earlier, Sen. Brian Schatz, D-Hawaii, urged Democrats to stop worrying about reaching a bipartisan deal, tweeting: "Regular people don't care whether we pass something with 51 or 60 votes.
"It's a pandemic and the largest economic contraction in 90 years," he added. "We must ignore those who call anything a Republican proposes a compromise, and anything a Democrat proposes partisan. We have to deliver."
In an interview on ABC News' "This Week," Sen. Bernie Sanders, I-Vt., said Democrats are more likely to work with Republicans in a bipartisan manner on other legislative ventures rather than dramatically alter course on the Covid-19 relief proposal.
"We all want bipartisanship. And I think you're going to see more of it as we move down the pipe," he said, pointing to infrastructure and lowering the cost of prescription drugs. "We are going to look forward to working with Republicans. But right now, this country faces an unprecedented set of crises.
"We have families who are watching this program right now who cannot feed their kids," he said. "We have millions of people who face eviction. We are in the midst of the worst pandemic in 100 years. We have got to act, and we have got to act now."