A source told the Daily Star newspaper: "R-Patz and Shannon have been getting really close. She's fancied him for years, but he's always been dating Kristen when their paths crossed at social events.
"She understands what Rob's going through. She split up with Andrew Garfield after he fell in love with his 'Spider-Man' co-star Emma Stone."
"Shannon's hoping they can be more than just friends."
Shannon's best pal Katy Perry recently enjoyed a friendly dinner date with Robert where she encouraged him to get out and have fun.
A source said: "Last Thursday night, Katy and Rob met up at the Soho House in West Hollywood for a bite to eat. They are close friends and can just be themselves when they are with each other.
"Katy has been trying to be as supportive as she can for Rob. She understands what he's going thru and wants to help him. Katy has been urging Rob to go out and be silly and have fun!"