The couple had parted ways before Brown was convicted of assaulting her in 2009. However, they are believed to be back together.
"No one close to them would be a bit shocked if they suddenly eloped. Rihanna has told her close friends point blank - I would marry Chris," contactmusic.com quoted a source as saying.
"She even says she would have never left him if the media scrutiny didn't get so bad. They're passionate, and when they're drunk, they've both made many comments about getting married," added the source.
News that the on and off couple could exchange wedding vows comes after it was reported that Brown has given his ex-girlfriend Karrueche Tran a generous payout following their split last week.
He broke up with Tran over his friendship with Rihanna and to avoid any bad feelings, he offered the model money so she can buy herself a house.
"Neither Rihanna nor Chris want to make things harder for her than they already are. Chris wants to set her up financially and do right by her - he wants her to stay in his life if possible," said a source.