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The number of members who broke rank is small enough that Democrats could justify boycotting the committee. But if Democrats don’t participate, it could leave high-profile Benghazi figures such as Susan E. Rice and Hillary Rodham Clinton — the would-be front-runner for the party’s 2016 presidential nomination — twisting in the wind if the committee subpoenas them to testify.

Their jeopardy could increase if Republicans win control of the Senate in November elections. Several Republicans have demanded that upper chamber institute its own inquiry or join the House.

But Senate Majority Leader Harry Reid, Nevada Democrat, said he is comfortable with the 25,000 pages of documents that the State Department has turned over, the various reports completed, and a State Department internal review that found several lower-level employees responsible for bad decision-making.

House Democrats scheduled a caucus meeting for Friday morning to determine their next steps.

Rep. Rosa L. DeLauro, Connecticut Democrat, circulated a letter to colleagues late Thursday floating the idea that Democrats appoint just one member to the committee, a move that would signify a sufficient protest to Republicans’ handling of the matter but also would give the party a voice in the process.

Conservatives, who sometimes have been at odds with Mr. Boehner, cheered the move and said the speaker likely got fed up with White House stonewalling of separate committee investigations.

“A lot of us had been asking for this for a long time,” said Rep. Raul R. Labrador, Idaho Republican. “I am really excited that Trey Gowdy is chairing that committee. He is just somebody who is a dogged seeker of the truth, and that is what we are looking for here. I think it is the right thing and I really praise the speaker for it, for making the right decision on that.”

Rep. Steve King, Iowa Republican, said that if Democrats oppose the makeup of the House committee, they could set up a similar panel in the Senate.

“They could set up their select committee in the Senate and run it just the way we are going to run one in the House,” Mr. King said.