The discharge petition process can be time-consuming and complicated, so Democrats who devised the strategy started early and carefully crafted their legislative vehicle. Insiders privately refer to the measure as a “Swiss Army knife” bill — one that was intended to be referred to every single House committee in order to keep open as many opportunities as possible for forcing it to the floor.

It would create a task force to help grandparents raising grandchildren, create a federal strategy for reducing earthquake risks, change the name of a law that governs stock trading by members of Congress, extend small business loans, protect veterans from the I.R.S., authorize a new Pentagon grant program to protect nonprofit organizations against terrorist attacks and more. Notably, the legislation was so broad and eclectic that it was referred to 20 committees, where it has sat idle for months. That was the point.

Mr. DeSaulnier’s intent was never to pass the elements of the bill, though he favors them all. It was to create what is known on Capitol Hill as a shell of a bill that would ultimately serve as the basis for a discharge petition — and a way out of the debt limit standoff.

“I wrote it in a way to be prepared,” said Mr. DeSaulnier, a former member of the Rules Committee who worked with Democratic procedural experts to craft legislation that could provide a debt-limit escape hatch. “I anticipated there would be these problems with the Republican caucus, whether it was abortion or the debt limit. I think it was the responsible thing as a legislator to do.”

Democrats say the beauty of the Mr. DeSaulnier’s bill — which Republicans have ignored — is that it long ago passed the threshold of being held in committee for at least 30 days, the minimum length of time to initiate a discharge petition to force action on legislation. And they said that the fact that it was under the jurisdiction of so many committees gave them several options for moving forward.