(Adds statement from Democratic congressman Gottheimer)

By Susan Cornwell, Richard Cowan and Jeff Mason

WASHINGTON, Oct 1 (Reuters) - In a rare visit to the U.S. Capitol, President Joe Biden tried on Friday to end a fight between the moderate and progressive wings of his Democratic Party that threatened to torpedo his domestic agenda.

Faced with a moderate faction that wanted an immediate vote on a $1 trillion infrastructure bill and a progressive arm that wanted to wait until there was agreement on a sweeping $3.5 trillion bill to bolster social spending and fight climate change, Biden sought to split the difference.

The former senator told his caucus during a 40-minute meeting that they could delay a vote on the smaller bill and sharply scale back the larger one to around $2 trillion. But his message that there was no rush belied the fact that Congress faces multiple approaching critical deadlines.

"It doesn't matter whether it's in six minutes, six days or in six weeks. We're going to get it done," Biden said.

A source familiar with Biden's remarks at the meeting said he told lawmakers, "Even a smaller bill can make historic investments."

The delay comes at a bad time for Congress, which has a lot of work ahead in the next few weeks.

The Treasury Department estimates that it has until about Oct. 18 for the government's $28.4 trillion borrowing limit to be raised by Congress or risk a debt default with potentially catastrophic economic consequences. Then on Dec. 3, the nation faces the risk of a government shutdown that could be politically damaging for Democrats.

By early next year, attention will focus on the midterm elections in November 2022, where history favors Republicans' chances of recapturing a majority in Congress.

House Democrats confirmed late Friday there would be no vote on the infrastructure bill that day.

'HAVE TO COME DOWN'

The most vocal opponents to the proposed $3.5 trillion size of the social and climate bill are moderate Senate Republicans Joe Manchin and Kyrsten Sinema, either of whom has the ability to prevent a bill from passing.

With the Senate divided 50-50 between the parties, Democrats hold the majority by virtue of Vice President Kamala Harris' tie-breaking votes. They aim to pass the larger bill without Republican support using a maneuver called "budgetary reconciliation."

House progressives acknowledged after the meeting with Biden that the $3.5 trillion number would need to be scaled back, though Biden agreed to their demand of passing the two bills in lock-step. Progressives fear that passing the smaller bill first would doom the larger one's chances.