The 'Papi' singer is said to have "freaked out" when Anthony suggested that her partner of two months start paying his way with three-year-old twins Max and Emme.
"Marc couldn't believe his ears when he heard Emme and Max utter the words 'Daddy Casper' during a recent visit," 24/7 Gossip quotes a source as saying. "He thinks Casper is nothing more than a young fling to make Jennifer feel better about herself after the divorce.
"Marc mockingly told Jennifer that if Casper wants to be called 'Daddy', he needs to start acting like it and pay some of their children's expenses instead of just sponging off of her! Jennifer freaked out and told Marc he was childish, but she promised to correct the situation."
Lopez has refused to confirm that she and Smart are dating, even though the couple were spotted on romantic trips to Hawaii and Morocco late last year.
The American Idol judge divorced Anthony in July after seven years of marriage.
> Jennifer Lopez on life after Marc Anthony: 'Loving myself is the key'