The 29-year-old 'Roar' hitmaker, who recently broke up with singer John Mayer, was recently seen spending time with Pattinson, 27, at the Coachella valley music and arts festival in California, reported Contactmusic.
Mutual friends of the duo are said to be rooting for them to get together.
"Katy and Rob are the best of friends with a very strong sexual chemistry. They care a lot about each other but have always been careful not to upset Kristen. But Kristen's off doing her own thing and accepting it's time for both of them to move on," said a source.
"Friends genuinely want to see them get it together properly. They think they bring out the best in each other," the source added.