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PLAISTOW, N.H. — Gov. John Kasich of Ohio needs every vote he can get to post a strong showing in the Republican primary here, crowded as it is with other so-called establishment governors, like himself, and the outsize popularity of Donald J. Trump.

But he appears to be drawing some enthusiastic supporters who may not ultimately prove very useful on Tuesday: Democrats who see him as the best bet in a Republican field they think is too extreme.

“I think he’s sensible,” said Peter Brumis, 71, a registered Democrat who attended a town-hall-style event Mr. Kasich held here on Monday.

Registered members of one party cannot vote in another party’s primary, so Mr. Brumis, a retired teacher who lives in Atkinson, N.H., said he was planning to vote for Hillary Clinton on Tuesday. But he said that he would vote for Mr. Kasich if he could.

These voters have popped up elsewhere in the state, like at another town-hall-style event last Thursday, when a Democrat took one of Mr. Kasich’s campaign signs home with her, hoping she could at least persuade her neighbors to vote for him.

Mr. Kasich has some qualities that appeal to liberal voters: He expanded Medicaid in Ohio, over the objections of Republican legislators, and he often speaks passionately on the trail of the need to help the poor and drug-addicted. He has made fun of hard-right proposals like abolishing the Internal Revenue Service. But he opposes abortion except in limited cases, supports the death penalty and frequently boasts of his budget-cutting efforts. A near-constant presence at Mr. Kasich’s events is a clock that shows the national debt ticking ever upward.

So when, on Monday, a reporter suggested he was “Democrats’ favorite Republican” and asked how he could “thread the needle” to appeal to more conservative Republicans, his response was curt.

“I don’t agree with you, is how we thread that needle,” Mr. Kasich said, and climbed into his campaign bus.

At Mr. Kasich’s next stop, in Windham, a sign was stuck in the snow: “Democrats for Kasich.”

Chuck Angelopulos, 76, had it printed in Massachusetts, where he lives, and brought it to New Hampshire in the hopes of swaying liberal-minded voters to Mr. Kasich. Independent voters can vote in either party’s primary.

“You’re better off having this guy in the event that a Republican wins,” said Mr. Angelopulos, who said he was a Democrat. “This guy is the best shot we have of having things get done.”

Even if Mr. Kasich is not promoting his support among some Democrats, his appeal to them could be a sign that a large number of independents will cast ballots for him in New Hampshire on Tuesday.

Some of those voters, interviewed at his campaign events last week, said they planned to do just that.

“It’s kind of like an insurance policy,” said Bruce McCracken, 66, a retired educator from Wolfeboro, who attended Mr. Kasich’s town hall in Alton.

Mr. McCracken said he was leaning toward supporting Mr. Kasich in the primary, even though he would most likely vote for a Democrat in the general election.

“If the Democrats don’t win, the Republican’s going to win. Which Republican do you want to be running the country?” Mr. McCracken said.

Bob Longabaugh, 85, of Alton Bay, N.H., who was deciding between Mr. Kasich and Senator Bernie Sanders, the democratic socialist from Vermont, said he was considering supporting Mr. Kasich for the same reason: If a Republican were to win in November, he would like it to be the Ohio governor.

“He’s the most moderate,” Mr. Longabaugh said.