It was after Monday evening's derby that the incident erupted.
According to Aftonbladet, which attended Mourinho's press conference after the derby, he is said to have stated that Galatasaray's bench "jumped like monkeys to get the guy a yellow card". And referred to a Fenerbahce player whom they wanted to get a yellow card.
Galatasaray has condemned the coach's words on its official X account.
"Mourinho has regularly used derogatory language about the Turkish people since he started working in Turkey, and now he has incorporated inhumane rhetoric into his immoral statements. We want to inform that we intend to report him to the police, and also report his racist statement to Uefa and Fifa".
Fenerbahce has also made a statement on X, defending their coach.
"A comment after the match from our coach has been deliberately taken out of context and distorted in a misleading way".
They write further that Mourinho's words only described the reaction from Galatasaray's bench.
"These comments cannot under any circumstances be associated with racism".
Mourinho took over as head coach at Fenerbahce in July last year and has previously been suspended for one match after criticizing Turkish football and match delegates.
The 62-year-old has coached major clubs such as Real Madrid, Inter, Roma, Chelsea, Tottenham, and Manchester United over the years. He has won two Champions League titles and multiple league titles.