The New York Times reported last month that Donald Trump Jr broached the possibility in a call to a senior Kasich adviser in May, telling him Trump's vice president would be responsible for domestic and foreign policy.
Kasich said in a CNN interview that he was never contacted directly but that one of his aides received the call from Donald Trump Jr and that the Times' account of the conversation was accurate.
"I'd be the worst vice president," he said. "I have too many opinions."
The Trump campaign ultimately chose Indiana Governor Mike Pence.
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Kasich declined to attend the Republican National Convention -- which was held in his own battleground state Ohio -- saying that it would have been "inappropriate" to attend without endorsing Trump.
"I wish that I could be fully enthusiastic. I can't be," Kasich said, declining to say who he planned to vote for in November.
"But I still think it's difficult if you are dividing to be able to win in Ohio.