The fashion designer and former child star has been spotted with the 'Moneyball' director numerous times in recent weeks and even accompanied him to Philip Seymour Hoffman's funeral in New York on February 7.
Bennett was close to Philip - who died of an apparent heroin overdose on February 3 - after directing him in 2005's 'Capote', for which he won the Oscar for Best Actor.
Ashley, 27, and Bennett, 47, have previously been spotted at Dan Tana's restaurant in Los Angeles in January, where he was working on editing his film, 'Foxcatcher'.
A source told UsMagazine.com: ''They were introduced and have started dating.''
Ashley is following in her twin sister Mary-Kate's footsteps by dating an older man, as she has been in a relationship with 44-year-old Olivier Sarkozy for nearly two years.
Ashley previously dated businessman David Schulte for around a year until the beginning of 2014 and previous to that she was with 'The Hangover' actor Justin Bartha between 2008 and 2011.
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