If the payroll tax cut were to extend from April through the end of the year, it could cost close to $700 billion, according to estimates from the Penn Wharton Budget Model at the University of Pennsylvania. Asked how such a measure would be financed, Mr. Kudlow told reporters at the White House that he did not yet have an answer.

“I think over time, we’ll make it up with much better economic growth,” he said.

Both Republicans and Democrats were cool to the idea. Senator Lindsey Graham, Republican of South Carolina and usually a staunch Trump ally, said the reaction among Republican senators had been “mixed,” adding, “I have to think about that.”

Senator Lisa Murkowski, Republican of Alaska, said she and other Republicans had questions about how a payroll tax would be paid for and how long it would last. She also raised concerns over the fiscal solvency of the Social Security Trust Fund, which is supported by payroll taxes.

“While it might have great stimulative effect,” Ms. Murkowski said, “I think we need to be looking at the overall impact.”

Democrats argued that Congress should focus on narrowly tailored measures that could be delivered quickly.

“We asked the economists what really can stimulate the economy fast, and there were things like food stamps — because people would spend that fast — unemployment insurance, refundable tax credits,” Ms. Pelosi said. “What we are doing has to be related to the coronavirus crisis.”

Trump administration officials, congressional aides, lawmakers and lobbyists involved in the stimulus discussions suggested on Tuesday that the negotiated package would be built on bipartisan agreement: protecting Americans who lose their jobs or are unable to work as a result of the coronavirus.