The Twilight Saga star has been spotted on several dates with the 22-year-old model in Los Angeles recently and friends believe he is completely smitten with her.
A source told People magazine that the couple have been dating "for a month or two" and the British actor, who split from Kristen Stewart almost five months ago after four years together, is besotted with Dylan.
An insider also told Life & Style magazine: "It's taken him a while to really get over Kristen but he knows she wasn't the right woman for him. Rob needs someone secure enough to handle the pressures of fame."
A second friend said: "Dylan is perfect for Rob because her family tries to live their lives out of the spotlight."
It was previously reported that Dylan's father Sean Penn has warned Robert, 26, not to disrespect his daughter.
A source recently said: "He gave his piece about not getting too heavy with his daughter if he's playing the field, as he doesn't want to see her hurt.
"But it's backfired. Rob backed off but Dylan found out and has gone tonto at her dad."
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