Rangers agreed a deal to sell the 15-year-old to the Premier League leaders last month.
But it was up to Gilmour and his parents to decide whether or not he should join Chelsea.
Rangers wonderkid Billy Gilmour has decided to join Chelsea
The Scottish giants had hoped to convince Gilmour to stay at Ibrox.
But now The Scottish Sun claim Gilmour and his family have indicated he will join Chelsea.
Chelsea line-up predicted to face Manchester United Wed, April 12, 2017
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Click through the gallery for the predicted Chelsea team to face Man United
Gilmour impressed on a trial with Chelsea earlier this season and can now officially make the move when he turns 16 on June 11.
He has trained with Rangers’ first team this season and was included in the squads for cup wins against Morton and Hamilton.
Rangers would have been due just £90,000 compensation if Gilmour had left Ibrox in the summer when he turns 16.
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