The couple were recently linked and while they have been trying to keep their relationship under wraps, Robert is said to be smitten with Elvis Presley's granddaughter.
A source said: "Rob and Riley are having a really great time together, event though it's very early. There is an attraction to each other. They just get each other.
"Rob thinks Riley has a cool level head and is not impressed by stardom at all. Rob just loves that about her."
The couple also enjoy spending time at Robert's house rather than taking in the Hollywood party scene and love making music together.
The insider added to America's OK! magazine: "He loves that she'll go into his kitchen for like 10 minutes and come back with a delicious plate of pasta. They just do typical stuff that has pretty much started all his relationships. They get into debates about politics, human rights, things they had read in the papers that morning.
"She's pretty good on the piano and he'll often pull out his guitar and they'll have a mini jam session, making up silly songs and funny limericks."
Robert recently split from on/off girlfriend Kristen Stewart, who is said to be devastated the heartthrob has moved on so quickly.