The pair have had several exchanges in the past as Morgan failed in his attempts to court Allen for his talkshows.
Quipping about her supposed link to The X Factor, Allen said: "Nice to know that i'm 80% confirmed to be a judge on xfactor, anyone thinking about running it by me ? (sic)."
Morgan responded: "Did Bianca Gascoigne turn it down then?"
Allen wrote in return: "Don't be a snob Piers, it doesn't suit you, oh wait a minute................carry on !"
Morgan then said: "'Laughter would be bereaved if snobbery died'. - Peter Ustinov. Anyway, if you need some tips on how to be a talent judge, you know where to find me. (Answer: Hollywood)."
Allen retorted: "YOU ARE SUCH A D**K !!!!!!!!!!!!!!!!"
Morgan responded: "I can always tell when I've got you riled, you start using capital letters. I'm off for a bike ride now, do try and calm down by the time I return...."
The talkshow host has previously described Allen as "a smart cookie", "good fun" and "a wild little rock chick".