The 'Stay' hitmaker, 26, and the 27-year-old rapper have been ''inseparable'' in recent weeks and she hopes he will ask her to start looking at properties soon.
A source told gossip website HollywoodLife.com: ''She'd love to maybe one day soon see if they can co-exist together and try the whole living with each other thing. But she's not trying to force anything. She wants it to unfold and to let him take control of that.''
The couple have already discussed the idea but haven't decided if they are ready to take their romance to the next level yet.
The insider added: ''It would be something that's new and fresh and something she'd like to give a shot but he's got to be on board. They've scrambled with the idea but haven't really come to a conclusion. It's all just a matter of time. There's no rush because time is on their side.''
The duo initially planned to keep their relationship casual, but quickly developed feelings for each other.
The source said: ''Rihanna's human and so is Drake and they both can't escape their humanity. It's a thought that they've been tossing around and we'll see what happens.''
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