His chief argument is that he is the only candidate in the race who won a statewide election in a state that President Trump carried. Mr. Bullock ran 25 percentage points ahead of Hillary Clinton in 2016, with the difference particularly notable in Montana’s rural areas. Hill County, along the Canadian border, backed Mr. Bullock and Mr. Trump each by 17 percentage points.

“The clincher for him is connecting with people,” said Tom Miller, the Iowa attorney general who is Mr. Bullock’s most prominent supporter outside his home state. “He connects with people better than anybody I’ve seen except Barack Obama and Bill Clinton.”

Mr. Bullock grew up in Helena, Montana’s capital, where he delivered newspapers to the governor’s residence in which he now lives with his three children. After law school he worked in Washington before returning home to Montana, where he became known for his fight for campaign finance transparency, which was featured in the 2018 documentary film “Dark Money.” He delayed the start of his presidential campaign until mid-May in order to shepherd a six-year extension of Montana’s Medicaid expansion through the state’s Republican-controlled Legislature.

His electability-and-results pitch echoes those of the former vice president — though Mr. Bullock’s stump speech makes only winking reference to Mr. Biden, who for decades represented Delaware in the Senate — and the presidential race’s other poll leader, Senator Bernie Sanders of Vermont.

“I think Montana can teach Washington, D.C., a lot,” Mr. Bullock told a few dozen people gathered in the living room of Joseph A. Foster, a former New Hampshire attorney general, in Nashua. “Some of you think, ‘Oh, Montana, it’s just a small state somewhere.’ There’s a greater population than states like Vermont or Delaware, I’m just throwing that out there.”

While Mr. Bullock’s political successes in Montana left him fluent in the language of working with Republicans, his first presidential campaign swing through New Hampshire revealed he was less adept at discussing some of the issues animating the Democratic Party’s progressive base.

In interviews and during question-and-answer sessions with Mr. Bullock, several voters volunteered that he was the only candidate who, in video interviews published by The New York Times last week, supported the death penalty in some cases.