The former hedge fund investor turned impeachment activist Tom Steyer, who entered the race a month ago and plans to spend millions of dollars of his own money to help fund his campaign, earned 3 percent in the Monmouth poll. It was his strongest finish in a debate-qualifying poll and puts him just one poll result shy of meeting the D.N.C.’s threshold. Though reaching 130,000 donors will be difficult in such a short amount of time, Mr. Steyer’s ability to spend large sums on advertisements could help him amass new donors relatively quickly.

Senator Kirsten Gillibrand of New York also enjoyed her best finish in a qualifying poll so far, earning 2 percent support in one for the first time since she officially kicked off her campaign in March. The poll result comes on the heels of a generally well-received debate performance for Ms. Gillibrand, which included a particularly memorable line: “The first thing that I’m going to do when I’m president is I’m going to Clorox the Oval Office.”

Ms. Gillibrand’s campaign recently announced that it had crossed the 100,000-donor mark, which puts her in striking distance of meeting one qualification benchmark. But in order to get on the stage in Houston, she will also need to earn 2 percent support in at least three more polls before the end of the month. (Qualifying polls can be conducted nationally or in early-voting primary states.)

Representative Tulsi Gabbard of Hawaii has crossed the 130,000-donor mark, but she, like Ms. Gillibrand, has only one qualifying poll so far.

Mr. Yang’s debate performance last week appeared to bolster his campaign’s fund-raising. A campaign spokesman said that in the four days after Mr. Yang debated, he raised more than $1 million from about 38,000 people, the vast majority of them new donors. By comparison, the Yang campaign raised $2.8 million over the three months ending June 30.

“The country heard my message and is ready to talk about real solutions to gun violence, the new realities of the American economy, and how we measure our health and success as a nation,” Mr. Yang said in a statement on Thursday. “I’m excited to have those conversations in Houston and throughout the 2020 election.”