Is this a mission impossible? That is uncertain, since Cruise, 56, has yet to comment about the challenge, according to Entertainment Weekly.
But if Cruise accepts, former MMA champion Conor McGregor said he is prepared to host a fight.
"If Tom Cruise is man enough to accept this challenge, McGregor Sports and Entertainment will host the bout," McGregor tweeted.
While it is unknown whether Bieber has a strong right hand, several celebrities -- including actress Lane Lynch and "Riverdale's" Cole Sprouse -- called him out for his writing hand, which penned a grammatical error in his tweet. Bieber wrote "your" instead of "you're."
According to the online betting site Odds Shark,, Bieber would be an early favorite to win at -200 to Cruise's +150, the Los Angeles Times reported.
It's certainly a baffling challenge, and some Twitter users are expressing their confusion and amusement.