Pattinson last year broke up with 23-year-old actress Kristen Stewart, his girlfriend of three years, after she publicly admitted to cheating on him with her married "Snow White and the Huntsman" director Rupert Sanders.
Pattinson and Stewart briefly reconciled, but officially called it quits four months ago.
A source close to the actor told People.com Pattinson, 27, and Penn, 22, have been "dating a month or two."
"He's crazy about her," the insider said.
Pattinson and Stewart co-starred in the blockbuster "Twilight" vampire film franchise.
Celebrity Families of 2014 [PHOTOS]