Sen. John Cornyn John CornynThe Hill's Campaign Report: GOP set to ask SCOTUS to limit mail-in voting Liberal super PAC launches ads targeting vulnerable GOP senators over SCOTUS fight Senate GOP faces pivotal moment on pick for Supreme Court MORE (R-Texas), a member of GOP leadership, quickly poured cold water Thursday on House Speaker Nancy Pelosi Nancy PelosiHoyer: House should vote on COVID-19 aid — with or without a bipartisan deal Ruth Bader Ginsburg lies in repose at Supreme Court McCarthy threatens motion to oust Pelosi if she moves forward with impeachment MORE's (D-Calif.) suggestion that state and local governments could need approximately $1 trillion more in federal assistance amid the coronavirus pandemic.

"That strikes me as a pretty outrageous number, just for state and local support," Cornyn told reporters.

He added that Congress had "already provided $150 billion" for state and local governments as part of last month's $2.2 trillion coronavirus package.

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Pelosi told reporters earlier Thursday that Democrats will push for including almost $1 trillion in the next coronavirus relief package to help states and local governments hit hard by the pandemic.

"We're not going to be able to cover all of it, but to the extent that we can keep the states and localities sustainable, that's our goal," Pelosi told reporters.

Whether Congress will provide more aid for state and local governments, and what strings will be attached to the funding, has emerged as an early flash point in the nascent talks over a "phase four" coronavirus bill, which would actually be the fifth package passed by lawmakers.

Cornyn told reporters on Thursday that he didn't think Congress is ready to talk about the "next step" on the legislation yet. But, he added, he would support allowing for states and local governments to have additional flexibility in how they use the $150 billion already signed off on by Congress, including allowing them to use it for revenue replacement as the coronavirus has shrunk their tax base.

"I think that's related to the coronavirus and I would favor giving them flexibility," he said.

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Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellFEC flags McConnell campaign over suspected accounting errors Poll: 59 percent think president elected in November should name next Supreme Court justice Mark Kelly: Arizona Senate race winner should be sworn in 'promptly' MORE (R-Ky.) sparked days of controversy when he told radio host Hugh Hewitt that he would support allowing states to declare bankruptcy, something they can't do under federal law.

"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," McConnell said at the time.

Asked about the remarks earlier this week, McConnell added that he was not indicating that states have to declare bankruptcy but that it was an "option" and that he wasn't "necessarily recommending it."

McConnell has also indicated that he is open to providing additional funding for state and local governments but has tied it to what he views as his "red line" for the upcoming negotiations: additional liability protections for businesses as they begin to reopen.

"My red line going forward on this bill is we need to provide protection, litigation protection, for those who have been on the front lines. ... We can't pass another bill unless we have liability protection," McConnell said during an interview on Fox News, calling the additional legal protections a "condition" for the bill.