The Premier League leaders look set to land one of the top young players on the planet after agreeing a £350,000 fee for the 15-year-old starlet.
Gilmour was expected to up sticks and move to London in the summer, but the new Gers boss has admitted that he last week made one final, concerted attempt to keep the teen in Glasgow.
"Billy is not only going to have a bright future in this club but he's going to have a bright future in football," said the Portuguese coach, who has already invited the Scotland Under-15 captain to train with his first team.
"He is a bright kid - I call him Billy the kid because of his age - and a clever kid for me is someone who anticipates everything that is going on before all the others.
"Billy is that sort of player. He understands what is going on and at only 15 years old he can compete with boys of 20 years old and has even come in to train with us in the first team.
"Now we have a situation with the club that he belongs to, with us, and a family situation.
"So we need to deal with it and to do that we have held a meeting with his parents. I won't share what was said at this moment but I did explain my point of view.
"I do believe that for Billy, the family ourselves or even Chelsea or another club later in the future to receive a player on another level, better ready to go on this adventure, it is better for him to stay here."
Gilmour must decide whether he wants to take a chance on the west London club's reputation for making stars out of talented youngsters, like Kevin de Bruyne and Romelu Lukaku, or whether a Chelsea switch would stunt his development in the same way another wonderkid, Martin Odegaard, has stalled since moving to Real Madrid.