In that conversation, first reported by The New York Times, Trump Jr. said that his father's vice president would be in charge of both foreign and domestic policy, leaving the real estate mogul at the helm of "making America great again,” according to the advisers.
Kasich’s camp rebuffed every attempt to get him on board, the advisers said.
Trump announced on Friday that he had selected Indiana Gov. Mike Pence as his running mate.
Trump said today he had never personally offered Kasich the chance to join the Republican ticket. The Kasich advisors also said an actual offer was never made.
In a statement to CNN, Trump Jr. threw doubt on the advisers' recounting of the conversation, although he did not outright deny its accuracy. "What am I a meathead?" he told CNN.
Kasich has said that that Trump must significantly change his rhetoric for him to endorse the billionaire. On July 4, Kasich’s chief strategist, John Weaver, called Trump’s campaign chairman, Paul Manafort, to let him know Kasich had no plans to endorse Trump before the convention, according to one Kasich adviser.
Even though Kasich has no plans to enter the convention hall today or on the convention’s final day, as governor of the convention's host state, he has kept busy around Cleveland. He has received twice-daily briefings from law enforcement agencies, met with the Mexican, British and Ukrainian ambassadors to the United States, spoken before several state delegations, participated in a panel on foreign policy and delivered remarks to the U.S. Hispanic Chamber of Commerce.
The Trump campaign did not immediately respond to a request for comment from ABC News.