Even before Mr. Biden ended negotiations on Tuesday with Senator Shelley Moore Capito, Republican of West Virginia, progressive Democrats had warned that Republicans were unlikely to embrace the scale of spending needed to address climate change.

Mr. Biden has now shifted his engagement to a bipartisan group of senators working on their own framework. While that group has not yet disclosed details, one of those senators, Lisa Murkowski, Republican of Alaska, said in an interview Wednesday that she was open to including some climate provisions.

“I think when you’re talking about infrastructure, it’s really easy — it’s important, actually — to talk about some of the things that allow for reduced emissions,” said Ms. Murkowski, who has helped to write climate legislation in the past. “When you’ve got upgraded pipeline, that’s a good thing. When you have efficiency with the new transportation system, that’s a good thing. Charging stations, E.V., is good.”

Senator Mitt Romney, Republican of Utah, said separately that the proposal had “a number of line items that relate to climate change” but acknowledged it would be more limited than what many Democrats are seeking.

But even if the group can agree on a plan that is palatable to Mr. Biden, it faces several obstacles, including questions about how it would be funded and whether it could attract the 60 votes needed to overcome a filibuster.

That is why Mr. Biden has also spoken to Senator Chuck Schumer of New York, the majority leader, about starting work on a budget blueprint that could allow Democrats to use a fast-track budget process and advance infrastructure legislation with a simple majority vote.

But that strategy could force Democrats to either modify or jettison key elements to ensure passage. And in order to pass that budget bill, leaders can afford little dissent — particularly in the Senate, where all 50 senators who caucus with the Democrats must remain united — which could lead to further changes in order to accommodate varying priorities.