Mr. Perez never aspired to be Democratic Party chairman. But weeks after Donald Trump beat Hillary Clinton, he was privately urged to do it by President Barack Obama, under whom he had served as labor secretary. Representative Joaquin Castro of Texas, among others, thought it was important to have a Latino at the head of the party. He joined other Latino officials who gave Mr. Perez a nudge during a late-night session at the Capitol Hill townhouse of Henry Muñoz, who was then the party’s finance chairman. Mr. Perez won an endorsement from former Vice President Joseph R. Biden Jr.

Now Mr. Perez presides over a much-maligned primary process that has led to a largely white December debate stage. And the rules he wrote have wound up benefiting his onetime D.N.C. chair opponent, the young man who won by losing in 2017: Mayor Pete Buttigieg of South Bend, Ind.

Since Friday, Mr. Perez has twice found himself the subject of coordinated pressure campaigns from the Democratic presidential candidates whose contest he is overseeing. First, the seven candidates who are set to debate Thursday threatened to boycott over a labor dispute at the university in Los Angeles where the event is scheduled to take place. Then, in a separate matter, nine candidates, spurred by the struggling Senator Cory Booker of New Jersey, called for the D.N.C. to lower qualification thresholds for the party’s debates in January and February.

Mr. Perez has been personally working the phones to elected officials and labor leaders trying to resolve the conflict in Los Angeles, but he has not been receptive to candidates who are upset about missing the debates. He has already decided, he said Wednesday, that for the January debate, candidates will be required to reach both donor and polling thresholds, not one or the other, as Mr. Booker’s letter requested.

He didn’t say if the thresholds would increase from the December requirements — 200,000 individual donors and at least 4 percent support in at least four qualifying polls, or 6 percent support in at least two early-state polls.