Kasich told CNN's Jake Tapper that he didn't receive a call himself. But he said one of his aides confirmed to him a New York Times report last month
saying Donald Trump Jr. tried to entice Kasich with a position as the most powerful vice president in history -- putting him in charge of all domestic and foreign policy -- was accurate.
"That's what one of them has told me, yes," Kasich told Tapper in an interview aired Sunday on "State of the Union."
The Trump campaign has previously denied the details of the Times' report. When it was published late last month, Trump campaign spokesman Jason Miller told CNN "it's completely ridiculous" and that an offer was never made. Trump Jr. also denied making such an offer.
The Ohio governor, however, told Tapper he was never interested in serving as vice president.
"I never considered it ... I'd be the worst vice president. I have too many opinions," said Kasich, the last Republican challenger left standing amid ahead of Trump's nomination.
Why he skipped the GOP convention
Kasich raised controversy in some GOP circles by declining to attend the Republican National Convention -- which was held in his home state -- but he's at peace with the decision, which he told Tapper was about "manners."
"If I wasn't prepared to go there and get up and endorse a nominee, I just thought it was inappropriate to go into that convention hall," Kasich said. "Some people are really furious with me about that. But I did what I thought I needed to do."
Added Kasich: "Believe it or not, I wanted to show respect to the nominee."
How Trump could earn his vote
Kasich said that response to his decision to not endorse Trump has been mixed and that he sometimes gets "a little grief over this."
"I've had a lot of people pound on me about you need to do this, you need to do this, this is about the party," Kasich said. "And I love my party, but I love my country. And I have to be true to myself. I wish that I could be fully enthusiastic. I can't be."
He refused to say whether he would vote for Trump.
"I don't know what's going to happen at the end," he told Tapper, but made it clear he will not be voting for Democratic nominee Hillary Clinton.
When asked if Trump could do anything to earn his vote, Kasich said: "There's so much water over the dam now, it's become increasingly difficult. But I want, you know, unifying."
'A little bizarre'
Kasich has long called for the party to unify, and expressed displeasure with Trump's hesitation to endorse several prominent establishment GOP candidates, notably House Speaker Paul Ryan, Arizona Sen. John McCain and New Hampshire Sen. Kelly Ayotte. (Trump eventually endorsed all three Friday night, several hours after the interview was taped.)
Kasich called the initial decision not to endorse "a little bizarre" and took issue with Trump's treatment of McCain.
"As far as I'm concerned, McCain shouldn't even have to run for election in the Senate," Kasich said. "He ought to be in the Senate as long as he wants to be."
To that end, Kasich said he would help McCain's re-election fight.
"I don't care what it takes," he said.
Can Trump win Ohio?
As for Trump winning the Buckeye State, Kasich isn't as optimistic, saying he believes the nominee will do well in the state but not well enough to carry it.
"He's going to win parts of Ohio where people are really hurting and where people of both parties have failed to fix our education system," he said. "But I still think it's difficult if you are dividing to be able to win in Ohio. I think it's really, really difficult."
The black hole
The Ohio governor also said he watched the Democratic convention speech delivered by Khizr Khan, the grieving father of a fallen Muslim US soldier, who attacked Trump from the dais, prompting Trump to engage in a multi-day feud with the Khan family. Kasich recalled his time spent comforting those killed overseas, saying he meets frequently with Ohio families who lose a child in combat and that is his job to comfort them.
"I've seen the black hole. I've had the deep mourning and the pain," Kasich said, referring to the loss of his parents, who were killed by a drunk driver. "But here's what I know: I believe the Scripture when it says those who give up their life, or serve someone else, will wear a big crown. That their service is marked in the book of life, never to be erased."