"It's hours or days, and I don't know when," the North Carolina representative told reporters in Washington.
Republicans in Congress have been pushing for spending cuts in exchange for raising the $31.4 trillion debt ceiling. McHenry said the list of issues still to be dealt with had been whittled down to just a few.
"We've had a long list for a long time. What I didn't anticipate is we'd have a very short list for a very long time," he said.
Treasury Secretary Janet Yellen announced on Friday that the deadline for congressional leaders to reach a compromise was pushed back to June 5. She warned that the US risked running out of cash to pay the bills if no agreement was reached.