The demands of those Democrats, members of a bipartisan coalition calling itself the Problem Solvers Caucus, put another wrinkle in Ms. Pelosi’s effort to win the speakership, a painstaking campaign she has undertaken in the wake of Democrats’ midterm election victories that has involved trying to put down a mini-rebellion among her longtime foes and newcomers demanding change. The preliminary test will be when Democrats meet on Wednesday to cast secret ballots for their leaders.

But the ultimate verdict will not come until January, when the new Congress must elect a speaker on the House floor. The threat of a leadership fight has given rank-and-file Democrats a tantalizing opportunity to air their grievances and make demands in the hopes of winning concessions from Ms. Pelosi, who has said she is certain she will have enough votes to reclaim the speaker’s gavel.

“For us, our rules proposal isn’t about changing leadership; it’s about changing a system that stymies the will of the common-sense majority,” the group said in a statement on Monday, adding that its proposals would “help spur immediate action on health care, immigration and infrastructure.”

The group is essentially asking Ms. Pelosi and the incoming Democratic leadership to fundamentally alter the procedures of the House, where the majority rules absolutely and its leaders have ultimate say on what legislative proposals are considered. It instead supports allowing bipartisan measures to have a chance of being debated and voted upon.

Among the group’s proposals is mandating that any bill receiving 290 co-sponsors would go to the House floor for debate and a timely vote, an idea that a senior Democratic aide said Ms. Pelosi had already agreed to include in proposed rules she will present when the new Congress convenes in January. Another would guarantee that any amendment that has at least 20 Democratic and 20 Republican co-sponsors would get a debate and a vote. The third request, aimed at empowering rank-and-file lawmakers, is that each member be allowed to introduce one bill, co-sponsored by a member of the other party, to be debated and considered on a committee on which he or she serves.

The effort has infuriated newly empowered liberals, who have accused the Problem Solvers Caucus of being “corporatist” Democrats beholden to the same political donors that Republicans rely on. They question why Democrats are always the ones to cede power when they attain it.

The nine Democrats are in addition to the group that signed the letter last week vowing not to support Ms. Pelosi, a coalition initially numbering 17 that has shrunk since its leaders first sought to form it. There are also at least four newly elected Democrats who did not sign the letter, but have said they will keep their campaign pledge not to back Ms. Pelosi. Another new member who said during his campaign that he would not back Ms. Pelosi, Representative-elect Gil Cisneros of California, added his name to the letter on Monday.