The Senate's Gang of Six on Tuesday offered a different plan that would reduce the deficit by $3.7 trillion over the next 10 years and increase tax revenues by $1 trillion by closing a variety of special tax breaks.

Reid also announced he would meet for negotiations once again at the White House on Wednesday, and added that he is well aware of the looming Aug. 2, deadline set by the Treasury.

“We're doing our very best, recognizing that we have 13 days left," said Reid.

The Senate adjourned just before 7 p.m. on Tuesday night and is set to return on Wednesday at 9:30 a.m.

At noon there will be a roll-call vote on an amendment to the underlying Veteran Affairs (VA) appropriations bill currently pending on the Senate floor.

—This post was updated at 7:38 p.m.