According to reports, the couple, both 60, were informed last month by ITV their usual Friday presenting slot on the daytime show was no more.
BBC sources told The Sun the couple would be offered their own gig on the rival station.
A TV source said: “Eamonn and Ruth have a huge fanbase and are very experienced.
"They can easily carry any number of shows, so producers at the Beeb are considering handing them a format to front.”
The BBC has been contacted for comment.
A spokesperson for This Morning said: “We have a stellar presenting line up on This Morning in Holly & Phillip and Eamonn & Ruth. Any additions to this line up will be announced in due course."
The Mail on Sunday is today reporting the couple fell out of favour with the This Morning team last year.
The paper reports the show’s editor Martin Frizell clashed with Mr Holmes in the run up to the December General Election after Scottish First Minister Nicola Sturgeon wanted to give the couple a Christmas tree decoration during her appearance on This Morning.
It was suggested Ms Sturgeon should receive a gift in return - but Mr Holmes objected saying that it broke strict impartiality rules for broadcasters in the weeks before an election.
Meanwhile, O’Leary appeared to speak out about his new gig in an interview earlier in the week.
In an interview with Ireland AM on Thursday, Dermot said of his potential new job "it is always tricky when there are colleagues and friends involved".
"There’s nothing confirmed yet, but you know, we enjoyed doing the summer, so we’d like to do more," he continued.
"But at the same time, you don’t want to tread on anyone’s toes or treat anyone badly, so you’re always sort of caught between a rock and a hard place on that sort of stuff."