The Hustlers star, 51, "has already contacted Jen to organize a sit-down, but it's going to take some persuading," a source tells OK!. "Jen's still a bit skeptical about Bennifer 2.0 and worries about the children being exposed to any unnecessary romantic dysfunction."
Garner, 49, graciously allowed Violet, 15, Seraphina, 12, and Samuel, 9, to spend time with Affleck's ex-girlfriend Ana de Armas, 33, only to see their whirlwind relationship fizzle out after less than a year.
“J.Lo’s willing to abide by their mom’s rules — and you can bet they’ll be tough!"
The "Let's Get Loud" singer certainly understands Garner's caution as a mother. She shares 13-year-old twins Max and Emme with her ex-husband Marc Anthony.
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"Marc trusts Jen’s decision to introduce Ben to their kids," the source commented. "Ben would love for J.Lo and his kids to meet, he thinks they’d definitely get along. But that all depends on Jennifer Garner. He’s not pushing it until everyone is comfortable with it, he knows better."
Lopez and Affleck — who dated for two years before calling off their engagement in 2004 — haven’t exactly been hiding their new-and-improved relationship. Since they rekindled their romance in April, former Gigi co-stars have been spotted out and about in Montana, Miami and Los Angeles.
"She's optimistic about her future with Ben," says the source.