Mr. Kasich’s volatility crosses party lines — and it has earned him a reputation for moodiness that has left even Republican allies treading lightly around him.

Jay Hottinger, a Republican state senator, summed up the challenge of working with Mr. Kasich in a recent text message to his wife. Mr. Hottinger, a veteran legislator, had attended one of the governor’s campaign stops in Ohio not knowing whether to expect warmth or chilliness from Mr. Kasich.

To his relief, Mr. Kasich acknowledged Mr. Hottinger’s presence at the event and embraced him afterward. “It was Dr. Jekyll today instead of Mr. Hyde,” Mr. Hottinger texted his wife afterward.

“Anyone who’s been around Governor Kasich knows that it depends on what kind of mood you catch him in,” Mr. Hottinger said.

Several of the governor’s longtime allies and adversaries said they were watching the newly genial Mr. Kasich, as a presidential candidate, with bewilderment.