"I can't imagine any kind of debt ceiling measure that could pass the Senate would also pass the House," McConnell told a Capitol Hill news conference.
However, any compromise measure that passed the House would be much more likely to pass in the Senate too, McConnell said.
The senator said the current impasse could only be solved by direct talks between President Joe Biden and newly-elected House Speaker Kevin McCarthy.
"The final solution lies between Speaker McCarthy and the president," McConnell said.
Many House Republicans have been strongly opposing efforts by the Biden administration to raise the national debt ceiling and are pushing for cuts in Medicare and Social Security, among other government spending programs, to stay within the current limits.
Earlier on Tuesday, Treasury Secretary Janet Yellen announced she was taking extraordinary measures to avert a major financial crisis if a compromise agreement was not reached.
Yellen sent a letter to congressional leaders in both parties warning that she was halting some reinvestment of government bonds in order to allow the government to continue operating at least until June as negotiations continue.