It was reported yesterday that the English Premier League leaders had won the race to sign the 15-year-old, who will be free to leave for a compensation fee of around £500,000 when he signs his first professional deal this summer.
While the move is yet to be finalised, Rangers fear the teenager could be about to accept Chelsea’s offer, which would mean a huge weekly wage packet once he turns 16 in June.
Caixinha has tried to persuade the player he’ll have greater chance of success long-term if he remains in Scotland.
The Portuguese head coach has told Gilmour he’ll be involved in the Ibrox first-team squad more regularly next season.
Several other big clubs, including Arsenal and Manchester United, have been monitoring the boyhood Rangers fan.