Cooper's office said Trump had called the governor and insisted on a full convention with no face masks or social distancing, and that Cooper expressed concern and suggested a reduced event.
Trump is now ready to accept the nomination at the city's owned VyStar Veterans Memorial Arena in Jacksonville, which houses 15,000 people. Republicans are required to hold a portion of the convention in Charlotte because of a contract signed by the party, but the celebration will take place in Jacksonville.
"The RNC is committed to holding a safe convention that fully complies with local health regulations in effect at the time," National Republican press secretary for the Republican National Committee, Mandi Merritt, said in a statement.
"The event is still two months away, and we are planning to offer health precautions including, but not limited to, temperature controls, available PPE, aggressive disinfection protocols, and available Covid-19 testing," said Merritt. "We have an excellent working relationship with local leadership in Jacksonville and the state of Florida, and we will continue to coordinate with them in the coming months."
White House press secretary Kayleigh McEnany said Monday that Trump, who has resisted wearing a mask in public, told her that "he has no problem with the masks."
"It is anyone's personal choice whether to wear a mask or not," McEnany said at a White House press conference. "He encourages people to make the decision that is best for their safety, but he told me that he has no problem with the masks and that he does what the local jurisdiction requires."
Florida Governor Ron DeSantis was asked Sunday by CNN's Natasha Chen if he assured Trump that the convention can take place, after DeSantis said residents should stay away from closed spaces, crowds and close contact, what he calls "the three C's". "
DeSantis replied: "It is a work in progress. We will try to reach a yes … Obviously we are in a dynamic situation."
"I think we'll be fine by then, with a couple of months to go, and we hope to see that."
CNN's Dan Merica contributed to this report.