Cruise has not, at this stage, responded to the challenge.
However, Conor McGregor has said that he would like to host the fight - and judging by the replies to Bieber's tweet, there are a lot of people who would like to see it happen.
"If Tom Cruise is man enough to accept this challenge, McGregor Sports and Entertainment will host the bout," McGregor wrote on Twitter.
"Does Cruise have the sprouts to fight, like he does in the movies? Stay tuned to find out!"
McGregor famously challenged Mark Wahlberg to a fight in April so maybe he was Bieber's inspiration.
"I've decided to challenge Hollywood actress Mark Wahlberg next. A fight for all the shares," McGregor posted in a series of tweets in April.
"I'll smack him and his 4 brothers around. It wouldn't be the first crew of brothers I've smacked around. Check the history books. The UFC history books."
That fight has still not happened either.
Most people on Twitter appear to think Cruise, who's 56 years old, would crush 25-year-old Bieber.