“One of our issues this past fall was we were treating legislation like a Christmas tree, and everyone’s favorite bauble got to get on the tree,” said Representative Elissa Slotkin, Democrat of Michigan, who welcomed Mr. Biden’s focus on fighting inflation, supply-chain and veterans issues. “What we heard last night, without using these exact words was, ‘My agenda is now more prioritized’ — and the priorities overlapped with what people are talking about in my district.”

On Capitol Hill on Wednesday, Mr. Manchin offered some new detail about how he would seek to narrow Mr. Biden’s agenda. In an interview with Politico, and later in comments to reporters, Mr. Manchin said Democrats should first raise revenue by undoing some of the 2017 Republican tax law and approving legislation to lower the cost of prescription drugs. From there, he said, the money should be used to both reduce the deficit and fund at least one major Democratic priority over a decade.

“Half of that money should be dedicated to fighting inflation and reducing the deficit, the other half you can pick for a 10-year program — whatever you think is the highest priority,” Mr. Manchin told reporters, noting that several of his colleagues wanted to focus on combating climate change.

“Everybody knows pretty much where I am,” he added. “If they’re not serious about inflation and debt, then you know, it’d be hard for me to negotiate.”