"Why would I feel compelled to support someone whose positions I kind of fundamentally disagree with?" he told Fox News.
Kasich suspended his own run for the White House a little more than a month ago and has since declined from endorsing Trump, decrying the divisiveness and "name-calling" he says has been a part of the campaign.
"Mr. Trump called me and said, 'What are you going to do to support me?' and I said, 'We're like two companies. We have a difference vision, a different value system, and a different objective,' So its pretty hard to put that together," Kasich told Fox News' Bill Hemmer, referencing a conversation that occurred a couple weeks ago.
The Republican National Convention is set to be held in Kasich's own state of Ohio in just over a month. But when asked if its possible he could walk into the convention in Cleveland and still not endorse Trump, Kasich said, "Absolutely. Of course."
Ohio, a swing state, will play a critical role in November's general election.
Kasich said he would not endorse Trump's main rival, presumptive Democratic nominee Hillary Clinton. But when asked about the choice between Clinton and Trump, Kasich said: "It's not that simple."
"I've said from the beginning that if somebody came into Ohio and they were a divider, that they couldn't win the state. It's not on me as to how Donald Trump does in Ohio. It's on him," Kasich said. "If you're going to insult Hispanics, if you're going to turn off minorities, if you're going to, you know, have a reckless suggestions on foreign policy, that's not good."
Kasich also cited the New York businessman's comments about the "Mexican" heritage of a federal judge overseeing a lawsuit on Trump University.
"He's trending the wrong way with me. Look at what just happened with this judge, here's a guy born in Indiana and this guy says that he's biased, or that if he was a Muslim he's biased. I mean that's not good stuff," Kasich said. "I just wish that there was a change. But it doesn't look like there's been any change. It looks like it's gotten worse."
Kasich also said there was never a consideration that he would run on a ticket with Trump, despite much speculation over a possible vice president position.
"A lot of people called and they talked to my folks and I've had people call me but was never ever ever any consideration and I didn't have that direct conversation with him," Kasich said. "We're too different."