AceShowbiz - Jennifer Garner seemingly doesn't want to have anything to do with Ben Affleck's new girlfriend Lindsay Shookus. And if she has a say on the matter, she would like to keep her and the actor's three kids away from the "Saturday Night Live" producer.

"Jen doesn't want the kids around Lindsay or having her in their lives," says a source to Us Weekly about the former spouses' kids, Samuel, Violet and Seraphina. "She has many reasons."

One of the reasons is rumors that "Ben had an affair with her," according to the source. Although the "Justice League" star and Lindsay went public with their relaionship in July last year, it was rumored that the pair started dating Lindsay back in 2014 when he was still with the "Love, Simon" star.

Ben, who split from Jennifer in 2015 after 10 years of marriage, allegedly "understands and has honored her request." The source adds, "They're good friends. She wants him to be honest and he usually is. She tends to find out about things, and Ben is very open with her. They both want what's best for the kids and have made a commitment to see that through."

"The one thing she's not flexible on is sobriety," the source continues saying. "Jen is hesitant to sign off until she's certain that the kids will be in the best hands at all time. Finalizing the divorce has been a long process because of it."

Ben and Jennifer, who filed for divorce last year, have been co-parenting their three kids. Just recently, they treated their kids to a Broadway outing. The family jetted to New York City to watch musical "Hello, Dolly!" on Thursday night, July 12.

The famous exes and their kids "were front and center at 'Hello, Dolly!' ", according to a so-called theather spy who told Page Six. They were "looking close as they watched Bernadette Peters and Victor Garber."

Another source added, "Jen and Ben both appeared to be really happy. They were laughing and smiling throughout the show and at intermission, and engaging with the kids. Some fans said hello and they were both really kind."