I think it's just republican trolling and stirring b/c Michelle has the princples and the guts to denounce and avoid tRump openly and the daily fail is a right wing, conservative, Tory/republican rag.
Obvs, I have no (f)actual information but based on their previous existence as reported frequently in media, it seems unlikely that JA and Obama would hook up. If you follow either of them even casually, neither seems like the other's type. Also, although I have no proof, both of them seem to have more common decency and integrity.
(For comparison, I can totally see Tony Blair and Wendy Ding having a few quickies.)