The "Umbrella" singer and Chris - who continue to insist they are just friends despite enjoying a romantic break in the Caribbean earlier this month - have been seen looking at luxury apartments in Los Angeles together.
A source revealed to America's Star magazine: "She and Chris want to live together. They play their music loud though, so they want a condo that's soundproof."
The couple are said to have visited several properties in Beverly Hills, including the luxurious Sierra Tower.
It has also been claimed Rihanna, 20, and 19-year-old Chris are engaged, after the Barbados-born beauty was seen wearing a diamond ring on her engagement finger.
Meanwhile, Rihanna partied with Leonardo DiCaprio on Tuesday night (August 26, 2008).
The "Disturbia" star began her evening at Hollywood nightspot Coco de Ville, before meeting Leonardo at the Villa lounge.
The stars and their entourages then went back to Leonardo's Hollywood home at around 2am to continue partying.
