Republican lawmakers unveiled a counterproposal on Thursday that includes a fraction of new spending on top of the expected reauthorization of current programs. President Biden will release his full $6 trillion budget proposal for the fiscal year 2022 on Friday.

Senate Republicans on Thursday proposed spending less that one seventh of what President Biden has requested in his expansive $1.7 trillion infrastructure initiative, countering with $257 billion in new funding for roads, bridges and other public works.

The narrow scope of the plan, which Republicans said would amount to a total of $928 billion over eight years when paired with existing programs — $1.4 trillion short of Mr. Biden’s proposal of new funds — illustrated the long odds that negotiations will yield a workable bipartisan compromise.

The latest Republican plan contains another probable deal breaker: They suggest paying for much of their proposal by repurposing funds from the $1.9 trillion pandemic relief law, an approach that White House officials have repeatedly rejected.

Instead, Mr. Biden has proposed large tax increases on corporations and wealthy taxpayers to pay for his much larger package, a prospect that Republicans, in turn, have refused even to consider.

“I haven’t had the chance to go over the details” of the counterproposal, Mr. Biden told reporters Thursday before boarding Air Force One to visit Cleveland.

The president said he had spoken with Senator Shelley Moore Capito of West Virginia, one of the lead authors of the Republican plan, and planned to meet with negotiators next week, adding that he wanted to see a bill “done” soon.

But his spokesman, Jen Psaki, while careful to praise the work of Ms. Capito and her allies, cast doubt on whether their comparatively small spending plan would ultimately pass muster.

White House officials “remain concerned that their plan still provides no substantial new funds for critical job-creating needs, such as fixing our veterans’ hospitals, building modern rail systems, repairing our transit systems, removing dangerous lead pipes, and powering America’s leadership in a job-creating clean energy economy, among other things,” Ms. Psaki said in a statement.

The White House also signaled a willingness to pursue talks into early June, when Congress returns from a Memorial Day recess, even as officials expressed concerns about the package’s narrow scope and its intention to repurpose pandemic relief funds.

The quartet of Republicans who proposed the latest plan includes Ms. Capito, Senators Patrick J. Toomey of Pennsylvania, Roy Blunt of Missouri and John Barrasso of Wyoming. The group said the proposal was proof their party was negotiating in good faith on an infrastructure deal.

They had initially presented a $568 billion plan for five years’ worth of overall spending, which also contained only a fraction of new spending; the outline presented on Thursday included about $70 billion more.

“We believe that this counteroffer delivers on what President Biden told us in the Oval Office,” Ms. Capito said, referring to a private meeting the senators attended with the president earlier this month. “It sticks to the core infrastructure features.”

Still, optimism for a bipartisan deal on infrastructure has dwindled despite an exchange of offers between the administration and Republicans, who have continued to object to Mr. Biden’s ambitions for the scope and size of a package. White House officials have expressed frustration with lawmakers’ reluctance to significantly increase the amount of new spending.

Several Democrats, wary of losing valuable time to act on their key priorities, are urging leaders to abandon the bipartisan talks and use the fast-track budget reconciliation process to advance the legislation, protecting it from a filibuster and allowing it to pass with a simple majority. A bipartisan group of senators is also quietly discussing their own proposal as a fallback option should talks between Republican senators and the White House collapse.

The Senate Environment and Public Works Committee also unanimously advanced on Wednesday a $304 billion reauthorization transportation bill, an effort that Ms. Capito said was “a major anchor” for a bipartisan accord.