The NBC/Wall Street Journal poll surveyed 426 Democrats from Feb. 14-17 and has a margin of error of plus or minus 4.75 percentage points.

The latest national polling average calculated by The New York Times, which was released late last week, put Mr. Bloomberg at 10 percent, behind Mr. Sanders, Mr. Biden and Ms. Warren.

Mr. Bloomberg formally entered the race in November, nearly a year after most of the other candidates. He failed to make the cut for the past several debates in part because he is not accepting outside contributions for his campaign. But new rules announced by the Democratic National Committee opened the door to his participation, as they enabled candidates to qualify for the Las Vegas debate, as well as the one that will take place on Feb. 25 in Charleston, S.C., without meeting a donor threshold.

He is certain to face onstage criticism Wednesday night from his rivals, especially Mr. Sanders and Ms. Warren, the two candidates who are not appearing at private fund-raisers and who have made targeting billionaires central to their campaigns.

Last week in Virginia, Ms. Warren told supporters that Mr. Bloomberg should not be the Democratic Party’s nominee because of his past remarks linking the end of redlining, a discriminatory housing practice, to the financial crisis.

Both Ms. Warren and Mr. Sanders previewed their arguments against Mr. Bloomberg in remarks on Tuesday.

“Anybody here with $60 billion, you can run for president, and you can buy the airwaves,” Mr. Sanders said in a speech in Reno, Nev. “That is called oligarchy, not democracy.”