Corker claims those cuts would result in $7.6 trillion in savings over a 10-year period.

Either way, Corker noted, most Republicans are going to demand significant decreases in government spending.

"I don’t think there is any Republican I know of that’s going to vote for a debt limit increase without dramatic changes in spending," said Corker.

The Commitment to American Prosperity Act, or the “CAP Act,” is co-sponsored by GOP Sens. Lamar AlexanderLamar AlexanderOvernight Regulation: Trump's Labor nominee hints at updating overtime rule Trump's Labor pick signals support for overtime pay hike Live coverage: Day three of Supreme Court nominee hearing MORE (Tenn.), Richard BurrRichard BurrSenate committee to question Kushner over Russian meetings: report Devin Nunes has jeopardized the oversight role of Congress Schumer: Trump must apologize for wiretapping claim MORE (N.C.), Saxby ChamblissSaxby ChamblissWyden hammers CIA chief over Senate spying Cruz is a liability Inside Paul Ryan’s brain trust MORE (Ga.), James InhofeJames InhofeRepeal of Obama drilling rule stalls in the Senate GOP senator: EPA 'brainwashing our kids' A guide to the committees: Senate MORE (Okla.), Johnny IsaksonJohnny IsaksonOvernight Finance: Senators spar over Wall Street at SEC pick's hearing | New CBO score for ObamaCare bill | Agency signs off on Trump DC hotel lease GOP senators offer bill to require spending cuts with debt-limit hikes Schumer to House GOP: 'Turn back before it's too late' MORE (Ga.), Mark KirkMark KirkObamaCare repeal bill would defund Planned Parenthood Leaked ObamaCare bill would defund Planned Parenthood GOP senator won't vote to defund Planned Parenthood MORE (Ill.) and John McCainJohn McCainSenate takes up NATO membership for Montenegro A great military requires greater spending than Trump has proposed Cheney: Russian election interference could be ‘act of war’ MORE (Ariz.).

Freshman Sen. Joe ManchinJoe ManchinPath to 60 narrows for Trump pick Overnight Healthcare: Ryan visits White House amid healthcare rubble Pence pushes Manchin in home state to support Gorsuch MORE (D-W.Va.) on Tuesday announced his support for the "CAP Act."