The co-chairmen of the Problem Solvers Caucus — Rep. Josh Gottheimer (D-N.J.), left, and Rep. Tom Reed (R-N.Y.) — flank President Trump at a Sept. 13, 2017, White House meeting on tax legislation in the Cabinet Room. (Jahi Chikwendiu/The Washington Post)

Nineteen House members say that, come January, they will not vote for any speaker candidate who does not agree to support a package of rules changes that could dramatically change how the chamber operates.

The 10 Democrats and nine Republicans making the pledge belong to the Problem Solvers Caucus, a group of centrist lawmakers who have sought to bridge partisan divides, and they say they will stick together and use their leverage to demand that the next speaker — whether it’s Kevin McCarthy (R-Calif.), Nancy Pelosi (D-Calif.) or someone else — endorse their reform proposals.

“We just want to see more progress being made,” said Rep. Brian Fitzpatrick (R-Pa.), one of the 19 who are going public Thursday. “I think everybody at their core knows that the way the rules are written need to change. They’re very archaic. They’re very antiquated. They don’t lend themselves to consensus-building and compromise. They don’t lend themselves to progress.”

The proposed reforms, first rolled out by the Problem Solvers in July, run the gamut from opening up the amendment process to making it easier for a bipartisan bill to get floor consideration to curbing the ability of a small group of rebellious lawmakers to force out a speaker. One proposal would force committees to deliver bills with more than 290 co-sponsors to the floor; another would require a three-fifths majority to pass any bill considered under a closed amendment process.

What the reforms have in common is that they would chip away at the power of the House majority party leadership, which has gained clout over the years at the expense of not only the minority, but also committee chairmen and rank-and-file members. Compelling a potential future speaker to relinquish some of that power could be a tough sell absent the kind of hardball tactics the Problem Solvers say that they are willing to employ.

Rep. Tom Reed (R-N.Y.), a co-chairman of the caucus, said that whether Republicans or Democrats are in control next year, they will probably have only a narrow majority, giving a small group of lawmakers leverage to demand reforms in exchange for their votes. “Congress is broken and my colleagues understand this — which is why we are gaining more and more interest and support every day,” he said.

A House speaker will be elected at the beginning of the next Congress in January by a majority of those present and voting — usually 218 in the 435-member House. A small group holding out for reforms could keep a candidate from meeting that threshold unless he or she submits to their demands.

Rep. Josh Gottheimer (D-N.J.), the other co-chairman, said the Problem Solvers’ proposals would make it easier to vote on “bipartisan ideas for the good of the American people” and pave the way for “common-sense legislation on everything from infrastructure to immigration.”

Many of the lawmakers are in tough reelection battles ahead of November’s midterms and might not return to Congress next year to vote on the next speaker, but they say the demand for a better-functioning House runs deeper than the Problem Solvers Caucus.

Republican moderates have griped for years about the hard-line tactics of the conservative House Freedom Caucus, which included threatening to depose then-House speaker John A. Boehner (R-Ohio). More recently, members of both parties have complained about an increasingly choreographed floor process that has made it harder for members to get bills brought up and amendments voted on.

Another one of the 19, Rep. Kurt Schrader (D-Ore.), acknowledged some irony in the fact that it might take a small group of centrists holding a speaker race hostage to get the reforms they are seeking. “But it’s for a good cause, not for some right-wing or left-wing ideology,” he said.

“There is a huge desire by the rank and file to get together and get stuff done,” Schrader added. “That means we’ve got to frankly work together on the floor, make sure that the speaker is the speaker of the whole House and not held hostage by a minority of folks like the Freedom Caucus or whoever. ... I think the American people would be better served at the end of the day.”

Besides Gottheimer and Schrader, the Democrats making the pledge are Reps. Jim Costa (Calif.), Thomas O’Halleran (Ariz.), Tom Suozzi (N.Y,), Salud Carbajal (Calif.), Daniel Lipinski (Ill.), Stephanie Murphy (Fla.), Vicente Gonzalez (Tex.) and Darren Soto (Fla.).

The Republicans joining them are Reed, Fitzpatrick, Fred Upton (Mich.), Lloyd Smucker (Pa.), John Katko (N.Y.), Carlos Curbelo (Fla.), Mike Gallagher (Wis.), Mike Coffman (Colo.) and Leonard Lance (N.J.).