“If they’re giving to my campaign, it’s because they believe in what we’re trying to do,” he said. “Look, I think the way politics ought to work is you explain what you’re for, and then people who believe in it will support you. I’ve never viewed this as something where you chase support with an idea.”

As South Bend’s mayor he bragged about “smart sewers” but rarely engaged in partisan food fights. When he ran for Indiana state treasurer in 2010, Mr. Buttigieg appealed to local Tea Party groups as a fellow traveler worried about government growing too large.

“There’s some, especially in my party, who think the Tea Party’s a wholly owned subsidiary of the Republican Party,” he said during an October 2010 candidate forum hosted by Citizens for Common Sense, a South Bend group affiliated with local Tea Party organizations. “But there are many others who believe that the Tea Party’s motivated by real concerns about the direction of our government, and the responses of our government to citizens.”

In Iowa, where polls show Mr. Buttigieg is in his strongest position — in third place behind Mr. Biden and Ms. Warren — he has been running TV advertisements since Labor Day in which he says “we need real solutions, not more polarization” and “others say it’s Medicare for all, or nothing.”

His shift to the center has come as political strategists see a potential new opening for a centrist, after Mr. Biden appeared wobbly in debates and recorded about one-third the cash on hand as did Mr. Buttigieg. But he’s also seeking to exploit the skepticism many Iowans have expressed about the single-payer proposals put forth by Ms. Warren and Mr. Sanders, according to Laura Hubka, the Democratic chairwoman in Howard County, Iowa.

“It seemed like to me he was more embracing Medicare for all and then as he went around through Iowa and talked to people, he got the idea that a lot of people are not comfortable with Medicare for all, people are scared,” said Ms. Hubka, the only county Democratic leader in the state to endorse Mr. Buttigieg. “Some of that is different from when he first came out.”

While Mr. Buttigieg’s shifts have left him vulnerable to attacks from progressive critics who view him as a donor-driven candidate without core convictions , they also come as he has elevated his standing in the historically crowded race.