One of Mr. Wiley’s first Ohio calls went to Jeff Polesovsky, the political director for Mr. Kasich’s presidential campaign and a senior aide during his two runs for governor. Mr. Polesovsky told Mr. Wiley that he isn’t ready to join the Trump campaign, according to people familiar with the conversation. Mr. Kasich, the last Trump rival to end his presidential bid, has declined to endorse Mr. Trump.

The governor hasn’t put out a formal notice to his aides to avoid working for Mr. Trump, but several of them interviewed said they wouldn’t take a job with him without first discussing it with Mr. Kasich, who, an aide said, didn’t wish to be interviewed.

“There’s no one in a senior- or midlevel position in the campaign in Ohio or in any of the states where we had staff who would or will be going to work for Donald Trump,” said John Weaver, Mr. Kasich’s senior strategist during his presidential campaign. “The very things that attracted them to John are the things that would keep them from working for Trump. Plus, they would be shot.”