The House is slated to vote Tuesday night on legislation stating that an increase in the federal debt ceiling would be acceptable only if immediate spending cuts are made, federal spending is brought back into balance over the next several years, and a balanced-budget amendment is approved by Congress.

“Before the week is out we could well be voting on the balanced-budget amendment as well, depending how long it takes to get through that process,” McConnell said.  

A senior GOP aide told The Hill on Tuesday that while they are “fighting” for a vote on the amendment for this week, there has been no word so far from Senate Majority Leader Harry ReidHarry Mason ReidBiden fails to break GOP 'fever' Nevada governor signs law making state first presidential primary Infighting grips Nevada Democrats ahead of midterms MORE (D-Nev.) on when that might occur. 

Reid, who controls the Senate schedule, slammed the balanced-budget amendment on Tuesday as "radical."

"Today the House will consider legislation that would force the nation to default on our financial obligations for the first time in history unless Congress adopts a radical, new constitutional amendment," Reid said from the Senate floor.