"He feels very hurt and expressed his feelings to Kim," the source says of Kanye. "Kim doesn't think there's anything wrong with Pete seeing the kids and wants all of them to make this work."
In addition to 6-year-old Saint, Kim and Kanye share North, 8, Chicago, 4, and Psalm, 3.
As for the kids, the source says that they "love Pete too and have so much fun with him."
"Kim waited to introduce them, but fully trusts Pete with them," the source says. "He really cares about her family and kids. She isn’t worried and she is happy that he wants to be part of their lives."
The source notes that Pete, who apparently got a tattoo of Kim and Kanye's kids' initials, "is willing to meet up with Kanye and speak to him if it will help."
"Pete fully has Kim’s back and will do whatever she wants and knows it is up to her," the source says of the 28-year-old comedian. "Pete just won’t put up with any crap from Kanye and doesn't want Kim to either."
Kim's family is also on board with the relationship, with the source telling ET that Pete "gets along with her sisters and Kris [Jenner] and Corey [Gamble]."
"They all love him and think Kim has never been happier," the source says of Kim's family. "They feel like Pete is just what Kim needed post-Kanye. Her friends and family tell her how they love seeing how happy she is. Kim is constantly in a good mood and smiling."
Meanwhile, 41-year-old Kim "is so happy and loves being with Pete," according to the source.
"Kim has laughed more with Pete than she has in years. It's changed her whole life," the source says. "She feels like he has been the best thing for her. She’s really just being in the moment and not too caught up in the future."
"They have amazing chemistry, he’s so sweet and there’s no drama. It’s refreshing. She thinks he is such a cool, good guy," the source adds. "Kim also loves how Pete can handle fame well between his own career and being in the public eye from his previous relationships. He is supportive of everything she does."
The source notes that Pete is equally "happy," and has been telling "his friends that Kim has become one of his best friends."
"He thinks she has such a good soul and that she is absolutely gorgeous," the source says of the former Saturday Night Live star. "He wasn’t expecting to ever date her and have such deep feelings for her. He feels like it's pretty wild in the best way."