The incident occurred at the end of the Turkish Cup quarter-finals on Wednesday, which Mourinho's side lost 2-1 at home.
In February, Galatasaray accused Mourinho of 'racist statements' after he said the home side's bench had been "jumping like monkeys" following a 0-0 Super Lig draw.
Fenerbahce said that Mourinho's comments were taken out of context.
While it was not clear which statements Galatasaray were referring to, Mourinho said the home side's bench had been "jumping like monkeys" and that the match would have been a disaster if a Turkish referee had been in charge.
Fenerbahce coach Jose Mourinho said the Video Assistant Referee went from being the invisible man to the most important man on the pitch after several controversial decisions went against his side in their 3-2 Super Lig win at Trabzonspor on Sunday.
Manchester United will travel to Turkish giants Fenerbahce, now coached by Jose Mourinho, among their eight games in the league phase of the new-look Europa League, the draw for which took place in Monaco on Friday.
The club's last league title dates back to 2014.
"Good evening Fenerbahce fans, see you tomorrow at Kadikoy (the Istanbul suburb where the club is based) and let's start our journey together," Mourinho announced.
Mourinho was 41 when he left Porto and took over at a Chelsea side poised to dominate English football following Roman Abramovich's takeover. Now Amorim, at 39, is the favourite to replace Jurgen Klopp at Liverpool.
Jose Mourinho threw his future as Roma manager into further doubt on Wednesday, saying he was "too tired" after the bitter pain of losing his first major European final to Sevilla.
Sevilla beat AS Roma 4-1 on penalties on Wednesday to win the Europa League for a record-extending seventh time, handing Jose Mourinho his first defeat in six European finals.
Roma manager Jose Mourinho has never lost a European final while Sevilla are six from six in Europa League finals. Something has to give in Budapest on Wednesday.
AS Roma coach Jose Mourinho said he was not fixated on becoming a legend at the club after leading his side to a second straight European final with a 1-0 aggregate win over Bayer Leverkusen after their Europa League semi-final, second leg on Thursday.
"The present is more important. The past is the past. We don't know the future," Mourinho said.
AS Roma grabbed a 1-0 home victory over Bayer Leverkusen in their Europa League semi-final first leg on Thursday thanks to 20-year-old Edoardo Bove's second-half goal.
AS Roma boss Jose Mourinho is keen on coming to Paris next season and replace Christophe Galtier after Luis Campos held talks with his agent.
Roma slid out of the Serie A top four by the slimmest of margins as they lost 3-1 to Atalanta in Bergamo on Monday.
Stephan El Shaarawy and Lorenzo Pellegrini scored in extra time to earn hosts AS Roma a spot in the Europa League semi-finals after defeating Dutch side Feyenoord 4-1 on Thursday for a 4-2 aggregate win.
Jose Mourinho brutally hit back at pundit Antonio Cassano following Roma's win over Torino after he criticised his team's style of play.
