The push for more state freedom in spending the first batch of money allocated by Congress last month comes as the clash intensifies over whether states should get hundreds of billions of dollars more in the next stage of emergency legislation — or whether there should be another recovery measure in the short term at all.

Senator Mitch McConnell, Republican of Kentucky and the majority leader, suggested last week that he would resist more state aid and said he wanted to push the “pause button” on new legislation. But on Monday, illustrating the pressure to help local governments, he seemed to backtrack in an interview on Fox News Radio, saying his initial comment was meant as a reminder that some states had fiscal issues before the pandemic.

“There probably will be another state and local funding bill, but we need to make sure that we achieve something that will go beyond simply sending out money,” Mr. McConnell said.

Other Republicans have joined with Mr. McConnell, saying they worry that states will take advantage of the opening to obtain federal money to resolve longstanding fiscal problems such as shortfalls in public employee pension funds.

“It’s not fair to the taxpayers of Florida,” Senator Rick Scott, Republican of Florida and a former governor of the state, told reporters in the Capitol on Monday. “We sit here, we live within our means, and then New York, Illinois, California and other states don’t. And we’re supposed to go bail them out? That’s not right.”