The D.N.C. spokeswoman Xochitl Hinojosa called the committee’s debate criteria fair, and noted that no campaign objected when the thresholds were announced earlier this year. No television network has agreed to host two nights of debates in January or February, she said.

“The D.N.C. will not change the threshold for any one candidate and will not revert to two consecutive nights with more than a dozen candidates,” Ms. Hinojosa said. “Our qualification criteria is extremely low and reflects where we are in the race.”

Mr. Booker failed to qualify for Thursday’s debate after failing to reach 4 percent support in any qualifying poll; four such polls were required. He said Wednesday that he would not “argue with the refs” about debate rules. Mr. Castro also has refrained from attacking Mr. Perez over debate rules.

It is not clear how robustly the other campaigns believe debate thresholds should be lowered. Some campaign staffers said privately Saturday that they felt obligated to sign the Booker letter or risk appearing racist, or unsupportive of an effort to be inclusive of candidates of color. Many have complained privately for months about the number of candidates on the debate stage, a gripe Mr. Biden has regularly made in public. In June and July, 20 candidates qualified for the debates, resulting in back-to-back nights with 10 candidates in each.

“Look, I think everybody knows these aren’t debates,” Mr. Biden told reporters during a September stop in Cedar Rapids, Iowa. “These are one-minute assertions.”

Mr. Perez, in an interview Wednesday, said next week’s debate field became only less diverse when Senator Kamala Harris of California, who had qualified, ended her campaign last week. No candidate, he said, has won the nomination without having at least 4 percent support by December.

“As we get closer and closer, you need to demonstrate that you are making progress,” Mr. Perez said. “That is a time-honored practice, and it’s a good practice because you do have to demonstrate that you’re making progress.”

Lisa Lerer contributed reporting.