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 AlexanderAndrew (Lamar) Lamar AlexanderThe risk of kicking higher ed reauthorization down the road Maternal deaths keep rising in US, raising scrutiny Supreme Court weighs future of online sales taxes MORE (Tenn.), Richard BurrRichard Mauze BurrOvernight Cybersecurity: Staff changes upend White House cyber team | Trump sends cyber war strategy to Congress | CIA pick to get hearing in May | Malware hits Facebook accounts Senators express concerns over Haspel's 'destruction of evidence' Overnight Cybersecurity: US, UK blame Russia for global cyberattacks | Top cyber official leaving White House | Zuckerberg to meet EU digital chief MORE (N.C.), Saxby ChamblissClarence (Saxby) Saxby ChamblissLobbying World Former GOP senator: Let Dems engage on healthcare bill OPINION: Left-wing politics will be the demise of the Democratic Party MORE (Ga.), James InhofeJames (Jim) Mountain InhofeDems give muted praise to Pompeo-Kim meeting Overnight Energy: Former Pruitt aide alleges more wasteful spending, retaliation | Senate confirms EPA No. 2 | Zinke backs off big park fee increases Senate approves Trump’s pick for No. 2 at EPA MORE (Okla.), Johnny IsaksonJohn (Johnny) Hardy IsaksonNow is the time to modernize the OTC monograph system Overnight Health Care: GOP pushes stiff work requirements for food stamps | Johnny Isakson opens up about family's tragic loss to opioids | Republicans refuse to back vulnerable Dem's opioids bill | Dems offer new public option plan The Hill's Morning Report: Haley clashes with White House MORE (Ga.), Mark KirkMark Steven KirkHigh stakes as Trump heads to Hill Five things to watch for at Trump-Senate GOP meeting Giffords, Scalise highlight party differences on guns MORE (Ill.) and John McCainJohn Sidney McCainTrump considering pardon for boxing legend after call from Sylvester Stallone GOP poised to advance rules change to speed up Trump nominees The Hill's Morning Report: Inside the Comey memos MORE (Ariz.).

Freshman Sen. Joe ManchinJoseph (Joe) ManchinPompeo lacks votes for positive vote on panel Democrats mull audacious play to block Pompeo Heitkamp becomes first Dem to back Pompeo for secretary of State MORE (D-W.Va.) on Tuesday announced his support for the "CAP Act."