Rangers struck a deal with the Premier League leaders last month, but they had hoped to convince him to stay put.
On contrary to this, the report states that Gilmour and his family have made it clear that he will join the Blues in the summer.
Gilmour, 15, is one of the brightest young talents to have come out from the club’s academy and Rangers will be deeply disappointed to lose him.
The youngster reportedly impressed on a trial spell with Chelsea earlier this season, and can officially make the move when he turns sixteen in June.
Chelsea have a star studded squad and as such, Gilmour will struggle to get immediately into the first team. In fact, it will be extremely difficult for him to break into the squad for the next three- to four years.
Moreover, there is a high chance that he could be sent on loan, and therefore a return to Ibrox on a loan deal in the future cannot be possibly ruled out.
Chelsea, meanwhile, are all set to win the Premier League title this season. Rangers, are third in the Scottish Premier League, and are aiming to grab the second position.