According to the Daily Star, the pair have been seeing each other for almost a month and are "absolutely smitten".
A source apparently said: "She's absolutely smitten with him and keeps saying he's the one. It's getting very serious very fast.
"She has been through a lot and Bruno is really making her feel special again. Who knows what will happen next? None of us would be surprised if they got engaged."
Last night, Mars won 'Favourite Male Artist In Pop/Rock' at the American Music Awards.
Meanwhile, Berrabah was forced to apologise after being convicted of assaulting a woman in a nightclub earlier this year.
Watch the music video for Mars's new single 'It Will Rain' below: