The 18-year-old is a Glasgow native who started out with the Old Firm giants before securing a switch to Stamford Bridge in 2017.
He had been training with the senior squad with Rangers since the age of 15 and was handed a competitive debut by Chelsea in August.
Big things are expected of him, with the Scotland Under-21 international not lacking in confidence or ability.
“He was 14 or 15 at the time and floating about Murray Park you'd see him playing and your eyes are just immediately drawn to him because he demands and his abilities are there to be seen.
“He's in the right positions, he makes the right passes and that's the thing that sets the top players apart from good players, and then good players from not so good players - decision making.
“His decision making is just...
“I genuinely say, and I don't mean any disrespect to anyone in the dressing room, but see when he came into training with the first team, his decision making was better than some players in our team.
“And that's not a slight on anyone, that's how good he is, at that age, at that level. The kid has got it.
“They won 7-0 one day and they had a decent team with Dapo (Mebude) and those two were the exceptional performers and probably a lot better than that level.
“They are sitting in the indoor hall as we are walking out that way to go on the bus so I pop my head in and ask the score.
“Wee Billy says, 'Seven nil'. I said, 'Oh right, decent game'. and he replies, 'Easy'.
“I said, 'Okay, brilliant. Who scored?' and he says, 'I scored'. 'You didn't score, you don't get in the box'. 'Aye I did, it was a penalty!'
“I walked out and thought, 'He's got character and personality. He's got a bit of baws about him'. And that is the huge, huge difference about him.”
Gilmour first figured for Chelsea off the bench in a Premier League clash with Sheffield United, before going on to make his full debut when handed a starting berth in a 7-1 Carabao Cup win over Grimsby.