According to Sky Italia's Gianluca Di Marzio, the purchase option stipulated in his loan agreement was activated after his appearance against Aston Villa (1-1) on Saturday.
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Bournemouth will pay AC Milan €19m plus up to €5m in bonuses to sign the Spaniard permanently.
Jiménez moved to England in the summer, and the wing-back has since made 22 competitive appearances for the Cherries and scored once.
The Spaniard is a former Real Madrid youth teamer.
