“If you can’t vote for a bill that says you’ll live within your means, then you’ve given up and you agree that the unsustainable path is the only one we have,” he added. “And that’s completely unacceptable."

Republicans are pushing hard to make the balanced-budget amendment part of any deal that would raise the nation’s debt ceiling but they have met stiff resistance from many Democrats. McConnell said that, by his count, he would need about 20 Senate Democrats to support it in order for it to clear the upper chamber. 

Amending the Constitution requires a two-thirds majority in both the House and Senate, and then ratification by three-fourths of the states.

“Every single Republican in the Senate supports a balanced-budget amendment,” he added from the Senate floor.

His comments come the day before the House is expected to vote on a “cut, cap and balance” bill, which is expected to pass. One of its requirements directs Congress to pass a balanced-budget amendment.

The "Cut, Cap, and Balance Act" is expected to fail in the Senate. President Obama issued a veto threat against it Monday.