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PAWLEYS ISLAND, S.C. — Gov. John Kasich of Ohio, edging toward more direct confrontation with former Gov. Jeb Bush of Florida, said on Thursday that Mr. Bush was risking tarnishing the Bush family name because of the negative tone of his campaign.

“I’m worried about Jeb,” Mr. Kasich said at a town-hall-style meeting at a pancake house here on South Carolina’s coast. “It’s all negative. How the heck can you sell negative?”

Speaking to reporters afterward, Mr. Kasich complained that Mr. Bush was spending his time “trashing people” and suggested that the Bush family’s reputation was at risk.

“I don’t know what he’s thinking,” Mr. Kasich said. “Does he realize the family legacy?”

This week, Mr. Bush criticized Mr. Kasich for expanding Medicaid in Ohio under the Affordable Care Act, and he also suggested that Mr. Kasich had been insufficiently committed to investing in the military.

“He’s not as conservative as me, and his record is not nearly as good as mine,” Mr. Bush said on MSNBC on Wednesday, while acknowledging that Mr. Kasich had done a good job as the governor of Ohio.

Mr. Kasich and Mr. Bush are competing, along with Senator Marco Rubio of Florida, to position themselves as the best establishment-aligned alternative to Donald J. Trump and Senator Ted Cruz of Texas.

With the exit of Gov. Chris Christie of New Jersey from the presidential race, Mr. Bush and Mr. Kasich are likely to engage in more direct conflict, as both men highlight their executive experience as they try to win over voters.

At the town hall, Mr. Kasich said he was sticking with the forward-looking style that helped his campaign stand out in New Hampshire, but added a warning: “I don’t take crap from anybody.”