Leaked footage of a private Romney fundraiser hit the press this week, featuring the candidate claiming that Letterman "hates" him for appearing on Jay Leno's Tonight Show.
"I've been on Letterman a couple of times, I've been on Leno more than a couple of times," Romney said to donors. "And now Letterman hates me because I've been on Leno more than him. They're very jealous of one another, as you know."
Letterman responded to Romney on Wednesday's edition of The Late Show by saying that Romney has an open invitation to appear on his series.
"Mitt [says] that I hate Mitt, that's what he says, he says that on the tape, because Mitt has done the Jay Leno show more often than he's done my show, and that we're all very jealous and petty and bitter and backbiting and showbusiness – unlike politics," Letterman quipped.
He continued: "I don't hate Mitt… And, I think now, more than ever, he and his lovely wife Mrs. Mitt are more than welcome to come on the show. And I'm telling you, if you think you're going to get to the White House, you've gotta spend time in this chair!"
Romney most recently guest starred on The Late Show with David Letterman in December 2011, a month prior to the start of the Republican primaries.
The candidate made his fourth appearance on The Tonight Show with Jay Leno in March. He sat down with Leno for the first time back in 2007.
US President Barack Obama was a guest on Letterman's programme on Tuesday (September 18).
Watch David Letterman speak about Mitt Romney below: