He has hit the ground running with three goals and three appearances for Jody Morris’ Under-18 side.
And Morris has been impressed with Gilmour on and off the pitch.
He said: “Billy is doing well.
“It is an adjustment when you come down to a club like Chelsea from Scotland.
“But he is adjusting really well and I’m really impressed with him.
“Not just as a footballer, but his personality and character is something that is standing out at the minute.
“The boy is not only a good kid, but someone who wants to learn and he wants to play at the very top.
“So he’s going to give himself the best chance if he carries on with that attitude.”
Morris added: “He’s certainly a talented boy and I’m enjoying working with him.
“He’s a boy that can play and get about the pitch and he wants to score goals.
“You can see that he has a good football brain.
“He listens and he takes on board information.
“That is just as important as all the stuff he does on the pitch.
“There are so many top players at our place, but he is certainly one that we’ve gone all out to get from Rangers.”
Chelsea beat a host of clubs to the signing of Gilmour, who featured on the first-team bench for the Gers before his move.
And Morris told The Sun: “I went to watch him a few times when we were looking at him, but at the same time you never really get to know someone until you see them close up.
“He has impressed our recruitment side of things and I have to say seeing him close up day in day out he has impressed me even more.
“But his personality and character is something that is going to benefit him because he gives it his all every day in training.”
