Vlasic was told by his new club that he could not wear the No 19 shirt when he arrived from Hadjuk Split, in a deal worth £8m, as it was reserved for Diego Costa. The Croatian midfielder eventually settled for the No 27 shirt.
Over the course of the transfer window, Costa was the subject of interest from various clubs, including AC Milan and Everton, after being frozen out at Chelsea.
However, the 29-year-old striker opted to return to former club Atletico Madrid, despite their transfer ban meaning he will be unable to play for them until January.
Vlasic spoke to Croatian website '24sata' and explain: "No 27 was free, I wanted 8 but Ross Barkley wears it, while the club was keeping 19 for Diego Costa."
Ronald Koeman decided to sign Vlasic after he impressed during a Europa League qualifier, and the 19-year-old admitted he was "sad" after Koeman was dismissed, but insists he is still "satisfied" at the Toffees.
"I'm sorry that Koeman got sacked, because he bought me and we had a really good relationship, but that's football," he added.
"Everton is a great club for young players like Jordan Pickford, Dominic Calvert-Lewin, Ademola Lookman, Tom Davies, me. We all get chances; I'm satisfied.