Speaking to reporters in Long Beach, Calif., on Saturday night during the state’s Democratic convention, Mr. Buttigieg said the just-released poll numbers were “extremely encouraging.”

“We have felt a lot of momentum on the ground,” he said.

His rise also suggests that voters, in Iowa at least, are increasingly favoring a centrist agenda — a view that has drawn two new entrants, Deval Patrick, the former governor of Massachusetts, and Michael R. Bloomberg, the former mayor of New York City, into the race this month.

The poll results reflect the deep split within the Democratic Party over whether it is veering too far to the left to defeat President Trump. Speaking on Friday to a room of wealthy liberal donors, former President Barack Obama expressed concern about some of the policy ideas being promoted by some of the candidates, citing health care and immigration as issues where the proposals may not align with public opinion.

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Though he did not single out any candidates directly, his remarks were seen as an implicit criticism of Mr. Sanders and Ms. Warren, two of the leading candidates who are pushing policy plans once considered too liberal, like “Medicare for all,” with the broader goal of “political revolution” and “big, structural change.”

“Even as we push the envelope and we are bold in our vision we also have to be rooted in reality,” Mr. Obama said. “The average American doesn’t think we have to completely tear down the system and remake it.”