"We have three critically things important before us right now that we need to do quickly," said Senate Majority Leader Mitch McConnell, who added that these are also the "top priorities" of the Trump administration.
To expedite approval of these must-pass items, Texas Sen. John Cornyn said he is "open" to adding an increase in the debt ceiling to a bill to provide immediate emergency aid to hurricane-ravaged Texas and Louisiana that is expected to move through the House and Senate this week.
"I would support that," Cornyn told CNN as he arrived back at Capitol from Texas, where he has been dealing with hurricane relief and clean-up efforts over the August recess.
Such a move could make it easier for GOP congressional leaders to get a politically fraught debt ceiling increase cleared before the end of the month when the US government could default on its debts.
In his remarks, McConnell did not directly address the issue of tying the debt ceiling to the disaster relief bill, something many conservatives oppose because they oppose increasing the debt ceiling unless under certain conditions and don't want to appear insensitive to hurricane victims by voting against emergency funds.
But the vocal support of Cornyn suggests the idea could get broad support from Republicans, although Cornyn, who is the GOP whip, said he had not had time to gauge all his colleagues.
Sen. Rand Paul, R-Kentucky, told CNN's Manu Raju he's prepared to battle the White House and Senate GOP over tying a debt limit increase to Harvey aid.
"I'll do anything to try to stop that," he said.
He said the White House effort to raise the debt ceiling is "like every other White House," which demands Congress raise the borrowing limit without any reforms.
Cornyn said if Paul filibusters a Harvey and debt ceiling bill, GOP leaders would vote to block Paul.
The Conservative Club for Growth also called on Tuesday for cuts in spending elsewhere to pay for Harvey aid.
"The idea of coupling an increase in the debt ceiling with a Harvey spending bill is equally abhorrent," Club for Growth President David McIntosh said in a statement. "For months, Secretary (Steven) Mnuchin advocated for a clean debt ceiling. That measure -- increasing our nation's spending ability without significant structural reforms -- was reckless enough on its own. But now, with the Senate's intention to add Harvey spending to a debt ceiling increase, even calling it a 'clean' increase is a misnomer; this action is dirtier than ever."
McIntosh's statement was a reference to comments Mnuchin made over the weekend, saying that in order to pay for hurricane relief, Congress will have to raise the debt ceiling.
"Without raising the debt limit, I'm not comfortable that we will get the money that we need this month to Texas to rebuild," Mnuchin told Fox News Sunday.
"That's our priority. We need to help the people in Texas, and we need to get that done."
Senate Democrats have suggested they are open to adding a debt increase to the Harvey bill but said they want to learn details of the measures before committing.
Cornyn said he didn't think a short-term government funding bill, known as a continuing resolution or CR, would be ready to also be added to the Hurricane Harvey relief bill so that would have to move separately later this month before the government runs out of money at the end of September.
"I'm open to the discussion on the (debt ceiling), I'm not sure we're ready to go on the CR yet," Cornyn said. "There is some urgency to get FEMA some additional funds, I'm told, by the end of the week. So, that would make it harder to do the CR but I do think it's fine to do the debt."
Cornyn said the debt increase would be clean and not include any measures to control future debt, something he otherwise supports.
"Unfortunately, I think that's all we have time to do. I'm told if we don't raise the debt ceiling than we can't appropriate the additional funds for Harvey on an emergency basis, which we absolutely need to do. I continue to be worried about the debt but I don't think this is the time to have that debate," he said.