His mother said Islam was told by the casting agent that he could not use his name if he wanted to be in the programme because "being called Islam if you are a boy is like a girl wearing the (Islamic) veil," Le Parisien newspaper reported.
The casting agents suggested Islam use "another Arab name" such as Mohammed or Sofiane. But Farah Alaouchiche refused.
The family left the premises to return to their home near Paris and never heard from the company again.
Angel Productions reportedly told the newspaper that "if Islam wasn't selected, it was not because of his name, but because there were more candidates than places available".
But the company did admit that "the casting agent did not react as she should have, there were words that hurt a little boy".
It promised to let Islam take part in a future edition of the programme, the paper said.
Fadela Amara, France's Minister for urban affairs, who is a Muslim, told Le Parisien she was "scandalised" by the incident, which she described as "disgusting".