As TMZ first reported, the LAPD is conducting a criminal battery and child endangerment investigation after Gabriel allegedly pushed the nanny with Nahla in her arms. And before the alleged pushing, the nanny says Gabriel screamed at her and hurled a racial epithet. In addition, we've learned the L.A. County Dept. of Children and Family Services is also conducting a child endangerment investigation.

TMZ has learned more about what nanny Alliance Kamdem has told the cops. Law enforcement sources tell us ... Kamdem claims Gabriel repeatedly yelled and cursed at her, often while Nahla was present. Kamdem also says it got so bad ... Gabriel would make her sit in a corner in his condo until he called for her.

Kamdem also claims Gabriel often yelled at Nahla, causing the child to cower and cry. And she told police... when Nahla had a rash he refused to treat it.

Kamdem quit last week after Gabriel allegedly shoved her.

Gabriel's rep tells TMZ, "Anyone can file a police report at any time, regardless of it being accurate."