Most forecasts at the moment anticipate that the party that captures the House will have a relatively narrow advantage, so corralling 218 votes for speaker will be no sure thing. That means a small group of lawmakers could form a bloc to control the outcome.

The Problem Solvers Caucus contains 48 members equally divided between Democrats and Republicans, probably more than enough to prevent the election of a speaker until demands for rules changes are met. A group of Republican reformers used the tactic in 1923 to loosen the leadership reins on the House.

“With narrow majorities on either side, there is a maximally favorable opportunity to seize the moment,” said William A. Galston, a senior fellow in governance at the Brookings Institution who first aired a similar plan five years ago as a way to resurrect bipartisan coalition-building.

The original idea by Mr. Galston and a colleague was to require the speaker to win the support of 60 percent of the membership, a fairly high bar. The group No Labels, through what it is calling the Speaker Project, is proposing that the number of votes needed to become speaker would be equal to the majority party’s full membership, in addition to five from the minority party. No speaker could serve without at least some minority support.

Members of the Problem Solvers Caucus have been meeting to review their ideas to see if they can reach a consensus on a package of rules changes. Together, the lawmakers and outside supporters hope to build momentum for the changes among colleagues, congressional candidates and voters heading into the midterm elections. They would then try to enact them either through party organizing meetings in November or via a floor fight in January.

“It seems like there is disruption brewing,” said Nancy Jacobson, founder and head of No Labels. “I feel like it is ripe.”