Romney referenced Letterman in now-infamous leaked footage from a private fundraiser, in which he said the Late Show host "hates" him for appearing more often on Jay Leno's Tonight Show.
Letterman responded by denying any ill-will towards Romney and invited the candidate on his Late Show, an offer which the Republican apparently hasn't accepted.
The CBS late-night host kept up his calls for a Romney interview earlier this week, jokingly telling his viewers that the election result hangs on the candidates' Late Show interviews.
"You can't blow this off, Mitt. I'm telling you, with every fibre of my being: if you want to go to the White House - not on the tour - you have to come here and shut me up," Letterman joked.
He went on to say: "You've gotta prove that I'm a dumbass punk and you've gotta come here and do it now... Because we wouldn't want anything to go the wrong way!"
Romney has made three appearances on The Late Show with David Letterman, most recently visiting in December 2011.
He has held a slew of media interviews in recent weeks on Good Morning America, Live with Kelly and Michael and 60 Minutes.