(CNN) — Former President Donald Trump said Saturday he would not debate Vice President Kamala Harris at all if she did not agree to attend a new Fox News debate next month, the latest in the public jockeying between the campaigns over when, if ever, the two candidates will meet on stage.

“I’ll see her on September 4th or, I won’t see her at all,” the former president wrote Saturday on his Truth Social platform.

Trump doubled down on that pronouncement at his rally Saturday evening in Georgia, telling attendees, “We’re doing one with Fox, if she shows up.”

“I don’t think she’s going to show up,” he added, claiming the vice president “can read a teleprompter” but can’t speak off the cuff.

Trump’s declaration comes after the former president on Friday night backed out of a planned September 10 debate hosted by ABC News. Trump committed to that event in May, after he and President Joe Biden agreed to move up the debate calendar and exchanged barbs about being willing to debate each other any time, any place.

Harris referenced those boasts on social media Saturday.

“It’s interesting how ‘any time, any place’ becomes ‘one specific time, one specific safe space,’” she wrote. “I’ll be there on September 10th, like he agreed to. I hope to see him there.”

A source familiar with the matter told CNN that it was the source’s understanding that ABC News would provide airtime to whichever candidate showed up – even if that ended up being just Harris. ABC News did not respond to CNN’s request for comment.

Trump’s call for a debate on his terms, with a “full arena audience,” continues the back-and-forth between the two campaigns as they try to shape the narrative around a potential debate, trading taunts and digs along the way. Trump has attacked his opponent’s intelligence, writing on his social media platform that Harris, who is also a former US senator and California attorney general, doesn’t have the “mental capacity” to debate him.

Harris’ campaign has argued that Trump is afraid to debate her.

Last month, Biden ended his reelection campaign and endorsed Harris after weeks of pressure from Democrats following his disastrous performance at a June 27 CNN debate. Trump has claimed without evidence that Democrats “staged a coup” to remove Biden from the ticket.

The public exchanges also call into question whether there will be a debate at all. Trump cruised to the Republican nomination earlier this year despite his refusal to appear at any of his party’s primary debates, arguing that he didn’t need to attend because he was a known entity and was leading in the polls. He has made similar remarks in recent days when asked about a debate with Harris.

The former president said Friday night that the ABC News debate could not happen because Biden was no longer the nominee and also because Trump is currently involved in litigation against the network.

Trump filed a defamation suit against ABC News and George Stephanopoulos in March after the host of “The Week” said the former president had been found guilty of “rape” in an interview with Republican Rep. Nancy Mace of South Carolina. Trump has been found liable for sexual assault and defamation of writer E. Jean Carroll. In January, a court awarded her $83.3 million in damages.

The ABC News lawsuit, however, was filed in March, months before the Trump and Biden campaigns agreed in mid-May on two debates.

The push for a Fox News debate marks the latest goal post-shifting from the Trump campaign since Biden stepped aside nearly two weeks ago.

On July 23, Trump told reporters on a press call that he hadn’t agreed to debate Harris, just Biden.

“But I want to debate her, and she’ll be no different, because they have the same policies,” he said. “I think debating is important for a presidential race, I really do, you sort of have an obligation to debate.”

Two days later, a spokesman for Trump’s campaign said it would not commit to any future debates until the Democratic Party formally announced a nominee, arguing that the party could still change its mind. At that point, Harris had secured pledges for a majority of delegates and no competitive Democrats had said they would run against her.

The Democratic National Committee announced Friday that Harris had secured enough support to win the party nomination a day after delegates began voting virtually to formally nominate a candidate.

On Monday, Trump told Fox News host Laura Ingraham he would “probably end up debating” Harris but said he could make the case for not showing up as well since they are both well known.

And on Friday morning, hours before he backed out of the ABC News event, Trump told Fox Business he wasn’t sure he wanted to debate Harris.

“I mean, right now I say, ‘Why should I do a debate?’ I’m leading in the polls, and everybody knows her. Everybody knows me,” he said.

Last year, Trump used similar arguments to explain his decision to skip the GOP primary debates, writing on his social media site that the public “knows who I am.”

Harris and her campaign have responded with taunts and calls for him to show up to the debate he committed to with Biden.

“He won’t debate, but he and his running mate sure have a lot to say about me,” Harris told a crowd in Atlanta earlier this week. “Well, Donald, I do hope you’ll reconsider to meet me on the debate stage, because as the saying goes, ‘If you’ve got something to say, say it to my face.’”

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