A source claimed the hookup with the actress and the director took place recently.
The magazine's report claims that Stewart was caught out meeting up with Sanders as they travelled around Los Angeles on July 17.
A witness said: "He was all over her. They would only take a break when they thought someone was walking by. It seemed like they couldn't get enough."
The pair reportedly hooked up after Stewart drove from the $6m Los Feliz home she shares with Pattinson.
A shutterbug followed the actress as she pulled into the parking lot of a deserted building, where Sanders arrived 15 minutes later.
The magazine (via RadarOnline) claims the director got into Stewart's car and they started kissing.
The witness said: "Kristen was sitting with her back up against the driver's-side window and this guy was kissing her entire body."
"Kristen is absolutely devastated," a source told People. "It was a mistake and a complete lapse in judgment. She wasn't having an affair with Rupert. It was just a fleeting moment that shouldn't have happened. She never meant to hurt anyone. She's a good person who just made a bad choice."
Hours before the story broke, Sanders' wife wrote a cryptic message to her Twitter followers.
"Sometimes good things fall apart so better things can fall together," she wrote, quoting Marilyn Monroe.
Shortly after following Us Weekly, she posted a tweet saying: "WOW".
She has since deleted her Twitter account, notes E! News.
Sanders has also issued a statement of his own.
"I am utterly distraught about the pain I have caused my family," the director said. "My beautiful wife and heavenly children are all I have in this world. I love them with all my heart. I am praying that we can get through this together."
Pattinson, 26, recently denied rumours that he and Stewart were planning on getting married.
"No one ever knows what is true or what isn't," he said in an interview with BlackBook magazine.
He has yet to comment on the scandal.
Us Weekly reports that the actor expressed his thoughts on infidelity in a recent interview.
In the April issue of Italian Vanity Fair, Pattinson was asked if he could ever date a married woman, he replied: "Life is not black and white. There are married couples that never see each other. Is that marriage?"
"But there's a thing I've never got: That is, why do people cheat?"
"I can understand the impulse," Pattinson continued. "But not how you can keep two relationships going at the same time for long."
He continued: "I'm not the casual-affair kind of guy. If I choose to be with someone, it's because I really want it. When I have a relationship, I'm 100 percent into it."
One of Pattinson's pals told Radar: "Rob is in shock that Kristen cheated on him, he is absolutely devastated.
"But right now he doesn't want to say anything in public until he figures out what he's going to do."
The pair were last spotted together at the 2012 Teen Choice Awards two weeks ago, while promoting the final installment of the Twilight franchise.
In June, Deadline.com reported that a Snow White sequel was in the works with both Stewart and Sanders.
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