Trump tweeted on Monday that he might have to pull the convention out of Charlotte if Cooper doesn't guarantee that full attendance will be allowed at the arena.
The conference is scheduled to start on August 24 at the Spectrum Center in Charlotte.
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Cooper was asked for his response to the tweets during a news conference on Tuesday.
"I'm not surprised by anything that I see on Twitter," he said. "It's ok for political conventions to be political, but pandemic response cannot be."
He said that the state health department has been in talks with the RNC about what the convention can look like.
ABC11 obtained on Tuesday a letter that North Carolina Department of Health and Human Services Secretary Dr. Mandy Cohen sent to the RNC in which she requested a written plan for how the convention plans to address COVID-19 safety protocols.
"Jordan Whichard from Governor Cooper's team shared with you the written protocols that NASCAR developed and then refined after discussions with our public health teams which allowed that event to occur in the Charlotte area this past weekend," she wrote. "While the RNC convention is obviously a very different event with its unique challenges for COVID-19, we hoped it would help illustrate the type of plan that would facilitate further conversations."
She continued: "The status of COVID-19 infections in our state and in the Charlotte area continues to rapidly evolve, thus, it will be important to have several scenarios planned that can be deployed depending on the public health situation."
"Everyone wants to get back into action soon, but I think everyone knows that we have to take certain steps to make sure we're protected," Cooper said on Tuesday.
He reiterated that the state wants to see a written plan from the RNC.
In the Rose Garden at the White House on Tuesday Trump said the Republican party needed "a fast decision" on whether the party could hold its convention with "the doors open" and that Cooper had "been acting very, very slowly and very suspiciously."
"We have to know before we spend millions and millions of dollars on an arena to make it magnificent for the convention and we have tremendous people," Trump said. "I mean, economic development consequences are tremendous for the state. We have to know that when the people come down they're going to have the doors open. Now, if the governor can't tell us very soon, unfortunately, we'll have no choice."
Currently, mass gatherings at venues like arenas are prohibited as part of Cooper's executive order because of the potential spread of the novel coronavirus.
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"I love the Great State of North Carolina, so much so that I insisted on having the Republican National Convention in Charlotte at the end of August. Unfortunately, Democrat Governor, @RoyCooperNC is still in Shutdown mood & unable to guarantee that by August we will be allowed...
...full attendance in the Arena," Trump wrote in a series of tweets the morning of Memorial Day.
MORE: @RepRichHudson, a staunch @NCGOP ally of @realDonaldTrump, remains positive & hopeful in statement to @ABC11_WTVD about @GOPconvention; cites @NC_Governor’s work to keep #CocaCola600 on schedule. @ABC @ABCPolitics @rickklein #abc11 #ncpol pic.twitter.com/aiL04GstTz— Jonah Kaplan (@KaplanABC11) May 25, 2020
The quadrennial conventions are among the biggest gatherings in the world and comparable only to the Super Bowl and Olympics when it comes to cost, publicity, attention and crowds.
Gov. Roy Cooper explains why North Carolina is not ready to reopen completely
"We are not going to sacrifice the health and safety of North Carolinians," Cooper said Tuesday. "That's the bottom line."