The Blues face competition from Chelsea for his signature.
The attacking midfielder has a £8.4 million release clause.
Arsenal are second favourites to sign Southampton's Virgil van Dijk and he would be the perfect acquisition.
Cook left Leeds for Bournemouth earlier this month.
The teenager starred during a 25-minute cameo performance in the win over Cambridge United on Saturday.
Celtic take on Astana in the Champions League qualifiers this afternoon.
The Gunners have lost Per Mertesacker to a long-term injury.
Nottingham Forest should consider signing ex-QPR man Alejandro Faurlin on a free transfer.
The Argentina defender has been a big hit since arriving at Goodison Park last summer.
Anton Walkes made his Spurs bow against Italian champions Juventus on Tuesday.