These criteria could easily halve the field: The first two sets of debates included 20 of the 24 candidates, but a New York Times analysis of polls and donor numbers shows that only 10 to 12 candidates are likely to make the third round.

[Which Democrats are leading the presidential race?]

Eight candidates have already met both qualification thresholds and are guaranteed a spot onstage. They are:

Former Vice President Joseph R. Biden Jr.

Senator Cory Booker of New Jersey

Mayor Pete Buttigieg of South Bend, Ind.

Senator Kamala Harris of California

Senator Amy Klobuchar of Minnesota

Former Representative Beto O’Rourke of Texas

Senator Bernie Sanders of Vermont

Senator Elizabeth Warren of Massachusetts

Ms. Klobuchar’s campaign announced on Friday that she had exceeded the required number of donors in the days following the debate. She had already met the polling threshold.

Two other candidates are very close: The former housing secretary Julián Castro and the entrepreneur Andrew Yang have surpassed 130,000 donors and each have three of the four qualifying polls they need. Representative Tulsi Gabbard of Hawaii has also crossed the 130,000-donor mark, her campaign said Friday, but she has only one qualifying poll so far.

Beyond them, only two candidates have even a single qualifying poll to their name: the impeachment activist Tom Steyer (two polls) and former Gov. John Hickenlooper of Colorado (one).

We asked their campaigns to provide donor numbers so we could assess where they stood. A spokesman for Mr. Steyer said he was “on track to collect the required number of donors to make the September debate stage” but did not give a number. Mr. Hickenlooper’s campaign did not respond, but Politico reported a month ago that he had only 13,000 donors.