Los Angeles, California - Former presidential candidate Robert F. Kennedy Jr. is reportedly fighting to hang on to his fraught marriage as he faces multiple adultery allegations.

A source close to Kennedy's wife, Cheryl Hines, recently told PEOPLE that the politician is desperately trying to salvage their relationship.

"She's been talking about filing for divorce, but he's begging her not to file,” the source told the outlet.

"But how many times can you forgive a partner that's dishonest and goes behind your back?" they added. "It's just ridiculous behavior from a grown, married man."

Politicians Tim Kaine backs Senate resolutions to block offensive weapons transfers to Israel

Last month, journalist Olivia Nuzzi (31) was placed on leave by New York Magazine after it was discovered she had a "personal relationship" – which was reportedly not sexual – with Kennedy while covering him on the campaign trail.

And last week, reports found that three women that work for an anti-vaccine advocacy group Kennedy chairs claimed they had affairs with him within the last year.

Sources have said that Hines is "thoroughly embarrassed and furious about his adulterous actions."

On Saturday, photos published by Page Six showed the couple driving around Malibu, California, the first time they've been seen publicly since the Nuzzi story broke.

A different source told PEOPLE that "things seem pretty tense" between the couple as they hung out over the weekend, and Hines was noticeably not wearing her wedding ring.