State and local governments facing dire financial straits because of the pandemic will have to wait until at least May before Congress considers further relief, Senate Majority Leader Mitch McConnell has indicated.

The House is set to vote Thursday on an interim round of coronavirus aid aimed at small businesses, and while Democrats sought to include roughly $150 billion in funding to shore up state and local budgets, the money didn't make it into the final bill because of objections from Republicans and the Trump administration.

After the Senate passed the bill Tuesday by voice vote, McConnell, a Kentucky Republican, predicted that future relief efforts would not be afforded such expeditious proceedings, citing concerns about the national debt and adding that "until we can begin to open up the economy, we can’t spend enough money to solve the problem."

He made clear that all lawmakers would have to be back in Washington before debating further rescue funds, including for state and local governments.

"I think the next debate, which I assume will relate to state and local government relief, needs to be when the Senate is back in session with full participation," McConnell said in response to a question about when he would be open to supporting the next phase of the funding Democrats had wanted to include in the interim package. "And in the meantime, also take a look at how much debt we've racked up and not try to wave something through the Senate, of that consequence, without full participation."

The Senate is not expected to return before May 4.

Across the country, state and local governments are clamoring for the federal government to rescue them from what could quickly become a fiscal catastrophe, saying that they may need as much as three-quarters of a trillion dollars as the pandemic dries up many of their revenue sources.

Democrats sought to include roughly $150 billion in funding to state and local governments in the aid package set for passage, though it was not ultimately included. Already, Congress approved $150 billion in funding for state and local governments as part of earlier coronavirus legislative aid — assistance governors and local leaders said would ultimately not be enough.

A Congressional Research Service report last week on initial coronavirus aid said that "early evidence suggests that the COVID-19 economic shock will have a notable impact on state and local budgets," pointing to the "sizable share of economic output" that derives from state and local governments.

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Bipartisan leadership of the National Governors Association called for $500 billion in aid to state governments to account for budget shortfalls, while counties and mayors have called for an additional $250 billion in emergency relief. On Monday, Sens. Bill Cassidy, R-La., and Bob Menendez, D-N.J., unveiled a legislative proposal for $500 billion in state and local funding.

Senate Minority Leader Chuck Schumer, D-N.Y., sounded optimistic Tuesday that such aid would make it into a future package, saying in a letter to senators that while he was disappointed the funding was not part of the upcoming bill, Democrats did secure a commitment from Treasury Secretary Steven Mnuchin "that he will support additional state and local relief in the next COVID-19 legislation, as well as a provision providing the flexibility to use all past and future relief dollars to offset lost revenues."

Following the Great Recession, state and local governments took years longer to recover than the federal government, contributing to a sustained drag on the economy.

The White House has acknowledged the bind these governments are in. White House senior adviser Kevin Hassett said Monday that a funding boost was on the president's "radar" and "if you look at state budgets, they're in about as bad a state as you've ever seen because the economy more or less ground to a halt."

On Tuesday, Trump tweeted, "After I sign this Bill, we will begin discussions on the next Legislative Initiative with fiscal relief to State/Local Governments for lost revenues from COVID-19," among other packages.

But McConnell reiterated Wednesday that any possible funding would come with strings — and that he was not in favor of bailing out states with more federal dollars.

"Fortunately, what they wanted to extract the most, I refused to go along with, and the White House backed me up, and that was we’re not ready to just send a blank check down to states and local governments to spend anyway they choose to," McConnell told conservative radio host Hugh Hewitt in an interview. "We had a tranche for them in the first bill, the $2.2 trillion bill. It has to be coronavirus-related. And I think we need to have a full debate not only about if we do state and local, how will they spend it."

Congress approved $150 billion in funding for state and local governments as part of earlier coronavirus legislative aid — assistance governors and local leaders said would ultimately not be enough. McConnell told Hewitt that he would favor states declaring bankruptcy.

"Yeah, I would certainly be in favor of allowing states to use the bankruptcy route," McConnell said. "It saves some cities. And there’s no good reason for it not to be available. My guess is their first choice would be for the federal government to borrow money from future generations to send it down to them now so they don’t have to do that. That’s not something I’m going to be in favor of."

Rep. Peter King, R-N.Y., blasted McConnell's statement on Twitter Wednesday night, calling the majority leader's "dismissive" remark "shameful and indefensible" and said, "To say that it is 'free money' to provide funds for cops, firefighters and healthcare workers makes McConnell the Marie Antoinette of the Senate."

.@senatemajldr McConnell’s dismissive remark that States devastated by Coronavirus should go bankrupt rather than get the federal assistance they need and deserve is shameful and indefensible. To say that it is “free money” to provide funds for... — Rep. Pete King (@RepPeteKing) April 23, 2020

...cops, firefighters and healthcare workers makes McConnell the Marie Antoinette of the Senate. — Rep. Pete King (@RepPeteKing) April 23, 2020

New York Gov. Andrew Cuomo said in an interview on WAMC Northeast Public Radio, "That is one of the saddest, really dumb comments of all time," when asked about McConnell's suggestion.

"Okay, let's have all the states declare bankruptcy — that's the way to bring the national economy back," Cuomo said sarcastically.

Sen. Rob Portman, R-Ohio, tweeted Wednesday that a "skyrocketing unemployment rate and subsequent decline in tax revenue has left local gov’ts stretched to the limit."

"As I told Ohio's county commissioners on our call this morning, I'm working w/ my colleagues to secure additional and more flexible funding," he said.

Trump, eager to restart the floundering economy by May and show that the U.S. has turned the corner on the pandemic, released guidelines for states to begin the reopening process even as some governors urged caution.

On Twitter and during daily White House press briefings, Trump has encouraged protesters demonstrating in some states against the strict stay-at-home orders put in place to slow the spread of the virus. Attorney General William Barr said Tuesday that the Justice Department would consider taking legal action against states where he believes stay-at-home orders are too restrictive.

"The idea that they are going to use the money to stay shuttered longer is ridiculous," Linda Bilmes, a professor at the Harvard Kennedy School and a leading expert on public budgetary and financing issues, told NBC News of one possible objection to further state aid. "I just can't emphasize enough how critical it is that the federal government should be back-stopping the local government."