Poyet has urged supporters to temper their expectations and also advised Gilmour to avoid falling into the same traps which previous Blues wonderkid McEachran fell into.
Speaking about Gilmour to Goal, Poyet said: “I think that it was important for the club to see him before the end of the season.
“He showed how mature he is especially in possession. Again, it isn't a matter of performing once or twice but every month, season through season.
“We should be happy to see him and hope he keeps developing his game slowly to become a better player because it is early doors. I really enjoyed watching him and he could be an interesting player for the future.
“I prefer to make sure people take time with him and not go the other way.
“I was at Stamford Bridge a long time ago when a young Josh McEachran was making the team at 17.
“The whole stadium was talking about him after that game. People were saying he was the next [Fernando] Redondo or [Pep] Guardiola.
“Unfortunately it didn't work out like that despite a great football career in the Championship. The idea to know is that it is difficult and [to know] how to progress in the next few months and years."