The 26-year-old is alleged to have been involved in a heated exchange with Song over his style of play during a training session on Saturday.
Song is said to have criticised Onana for taking too many risks with his feet near his own goal and he wanted the Inter Milan shot-stopper to play more direct.
Onana, who played the full 90 minutes in the 1-0 defeat to Switzerland last Thursday, allegedly refused to change his style of play and he has since been removed from Cameroon's World Cup squad in Qatar.
Cameroon's number one was not involved in the thrilling 3-3 draw against Serbia on Monday, with Devis Epassy starting in his place between the sticks.
After the match, Song confirmed that Onana has left the squad due to "disciplinary reasons" but the goalkeeper has been asked to return to discuss his situation.
Speaking at a post-match press conference after the draw with Serbia, Song said: "Onana is not in the Cameroon team for disciplinary reasons.
"He asked to leave and we accepted it after telling him on Sunday that he was going to be a substitute."
Song added: "He is a very important player but we are at the World Cup and my role is to put the team first ahead of an individual.
"He is one of the best keepers in Europe. This is not about his performances but you have to preserve the squad.
"Maybe it was the trigger we needed for this performance (against Serbia). It was a big risk that I took, but I'm the dad of these kids so when I have to take risks and make decisions, I do that and I stand by what I do.
"At the moment I've asked Onana to wait and we will see if he is going to stay with us. It's up to him."
Onana joined Inter Milan from Ajax on a free transfer over the summer and has since displaced Samir Handanovic as their new number one this season.
The former Barcelona youngster has kept five clean sheets in 13 appearances for the Serie A giants, while he has kept 14 shutouts in 34 international appearances for Cameroon.
Jean-Charles Castelletto, Vincent Aboubakar and Eric Maxim Choupo-Moting were all on the scoresheet as Cameroon claimed a valuable point against Serbia, keeping their slim hopes of progressing to the last 16 alive.
The Indomitable Lions will now prepare for their final Group G fixture against five-time world champions Brazil at Lusail Stadium on Friday.