In challenging Biden to fulfill his pledge of unity, the group said in a letter that their counterproposal will include $160 billion for vaccines, testing, treatment and personal protective equipment and will call for more targeted relief than Biden's plan to issue $1,400 stimulus checks for most Americans.
Winning the support of 10 Republicans would be significant for Biden in the 50-50 Senate where Vice President Kamala Harris is the tie-breaker. If all Democrats were to back an eventual compromise bill, the legislation would reach the 60-vote threshold necessary to overcome potential blocking efforts and pass under regular Senate procedures.
"In the spirit of bipartisanship and unity, we have developed a COVID-19 relief framework that builds on prior COVID assistance laws, all of which passed with bipartisan support," the Republican senators wrote. "Our proposal reflects many of your stated priorities, and with your support, we believe that this plan could be approved quickly by Congress with bipartisan support."
The plea for Biden to give bipartisan negotiations more time comes as the president has shown signs of impatience as the more liberal wing of his party considers passing the relief package through a process known as budget reconciliation. That would allow the bill to advance with only the backing of his Democratic majority.
The Republicans did not provide many details of their proposal. One of the signatories, Louisiana Sen. Bill Cassidy, said that it would cost about $600 billion.
"If you can't find bipartisan compromise on COVID-19, I don't know where you can find it," said Ohio Sen. Rob Portman, who also signed the letter.
The other GOP senators are Susan Collins of Maine, Lisa Murkowski of Alaska, Mitt Romney of Utah, Shelley Moore Capito of West Virginia, Todd Young of Indiana, Jerry Moran of Kansas, Mike Rounds of South Dakota, and Thom Tillis of North Carolina.
Brian Deese, the top White House economic adviser who is leading the administration's outreach to Congress, said administration officials were reviewing the letter. He did not immediately commit to a Biden meeting with the lawmakers.
But Cedric Richmond, a senior Biden adviser, said the president "is very willing to meet with anyone to advance the agenda." When asked about the senators' plan, Richmond said, "this is about seriousness of purpose."
Deese indicated the White House could be open to negotiating on further limiting who would receive stimulus checks. Portman suggested the checks should go to individuals who make no more than $50,000 per year and families capped at $100,000 per year.
Under the Biden plan, families with incomes up to $300,000 could receive some stimulus money.
"That is certainly a place that we're willing to sit down and think about, are there ways to make the entire package more effective?" Deese said.
As a candidate, Biden predicted his decades in the Senate and his eight years as Barack Obama's vice president gave him credibility as a deal-maker and would help him bring Republicans and Democrats to consensus on the most important matters facing the country.
But less than two weeks into his presidency, Biden showed frustration with the pace of negotiations for relief at a time further signs of economic wear from the pandemic. Last week, 847,000 Americans applied for unemployment benefits, a sign that layoffs remain high as the coronavirus pandemic continues to rage.
"I support passing COVID relief with support from Republicans if we can get it. But the COVID relief has to pass - no ifs, ands or buts," Biden said on Friday.
In the letter, the Republican lawmakers reminded Biden that in his inaugural address, he proclaimed that the challenges facing the nation require "the most elusive of things in a democracy: Unity."
Cassidy separately criticized the current Biden plan as "chock-full of handouts and payoffs to Democratic constituency groups."
"You want the patina of bipartisanship ... so that's not unity," Cassidy said.
Jared Bernstein, a member of the White House Council of Economic Advisers, said Biden remains willing to negotiate but that officials needed to see more details from Republicans. At the same time, Bernstein pressed the administration's argument that doing too little to stimulate the economy could have enormous impact on the economy in the near- and long-term.
"Look, the American people really couldn't care less about budget process, whether it's regular order, bipartisanship, whether it's filibuster, whether it's reconciliation," Bernstein said. "They need relief, and they need it now."
Portman and Deese were on CNN's "State of the Union," and Deese also was interviewed on NBC's "Meet the Press." Cassidy and Bernstein appeared on "Fox News Sunday" and Richmond was on CBS' "Face the Nation."