Nobody else is close to qualifying. Senator Cory Booker of New Jersey and the former housing secretary Julián Castro have met the donor threshold but have no qualifying polls. They each had 1 percent support in the Quinnipiac poll. In a national Monmouth University poll released earlier on Tuesday, Mr. Booker had 2 percent support and Mr. Castro had 1 percent.

In terms of the overall contours of the race, the polls showed little change. Mr. Biden retained a clear lead, with Mr. Sanders in second place, Ms. Warren in third and Mr. Buttigieg in fourth. (In the Quinnipiac poll, the difference between Mr. Sanders and Ms. Warren was within the margin of error.)

Notably, former Mayor Michael R. Bloomberg of New York, who entered the race less than three weeks ago, had 5 percent support in both polls. Mr. Bloomberg has already spent tens of millions of dollars of his own money on advertising and has not been seeking donors, a decision that will exclude him from debates regardless of his poll numbers unless the D.N.C. drops its donor requirement.

Also working against Mr. Bloomberg: Voters who know of him tend not to like him. In both polls released Tuesday, he had the worst favorability to unfavorability ratio of any Democratic candidate for whom that question was asked.