"We need to assess what we've already done, take a look at what worked and what didn't work, and we'll discuss the way forward in the next couple of weeks," Senate Majority Leader Mitch McConnell told reporters after President Donald Trump spoke to a Senate Republican luncheon.
House Republican leader Kevin McCarthy also said he saw no need for another coronavirus bill now, but told reporters he could back lengthening the loan period for small businesses to 10 or 12 weeks from the current eight.
Senate Small Business Committee Chairman Marco Rubio said he hoped legislation could pass this week to alter the loan program helping restaurants and other small enterprises survive prolonged closings during the pandemic.
Such a measure would give businesses more time to spend money they receive under the loans, which can turn into grants.
McConnell is under pressure from the Democratic-controlled House, which passed a new, wide-ranging coronavirus bill on Friday. Its $3 trillion price tag is about equal to the cost of four previous measures enacted by Congress.
Some of McConnell's fellow Republicans are clamoring for new emergency measures to respond to a pandemic that has killed more than 90,000 Americans and taken a heavy toll on the economy.
A bipartisan group of senators on Monday signed onto a bill to establish a $500 billion fund to help state and local governments, whose revenues are severely curtailed by coronavirus-related closings.
Some lawmakers also have discussed the need to revise unemployment insurance provisions with the national jobless rate spiking to 14.7 percent in April.