Sen. Corker: White House is running orchestrated campaign against CAP Act

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 Alexander (Tenn.), Richard Burr (N.C.), Saxby Chambliss (Ga.), James Inhofe (Okla.), Johnny Isakson (Ga.), Mark Kirk (Ill.) and John McCain (Ariz.).

Freshman Sen. Joe Manchin (D-W.Va.) on Tuesday announced his support for the "CAP Act."