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.
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