"We're going to take a pause here, we're going to wait ... and clearly weigh before we provide assistance to states and local governments," McConnell said Wednesday.
Republicans are not ready to send a "blank check" to the states, McConnell said. "We all have governors regardless of party who would love to have free money," he said.
McConnell said he sees "no good reason" why laws shouldn't be changed to allow states to enter into bankruptcy proceedings, which they are now unable to do.
Such a suggestion is highly unlikely and governors delivered swift blowback.
Gov. Andrew Cuomo, who has warned New York will lose anywhere from $10 billion to $15 billion in revenues during the epidemic, called it "one of the saddest, really dumb comments of all time."
In a radio appearance Wednesday, Cuomo said: "You want to reopen the economy, Mr. McConnell, so everyone gets their job back? But the people you put in charge of reopening, the governors and the states, should declare bankruptcy?"
New Jersey Gov. Phil Murphy said: "Really? This is the time, in a moment of crisis unlike any our country has faced in at least 100 years, to suggest it's a good thing for states to go bankrupt?"
"Come on, man. That is completely and utterly irresponsible," Murphy said.
Murphy calls McConnell's comments 'utterly irresponsible'
Trump has said he supports including fiscal relief for state and local government in another virus aid package along with infrastructure projects. It's unclear how soon the next bill can advance. There's also the question of when Pelosi and McConnell feel comfortable reopening Capitol Hill, though Trump has signaled he wants discussions to begin as soon as Congress finishes the current legislation.
The Senate is scheduled to return May 4.
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Information from ABC News and the Associated Press
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