The Arsenal ace is now in the final 12 months of his current contract and has been tipped to leave the Emirates this summer.
Manchester City and Liverpool have been linked with raids for the 23-year-old, while Starsport understands Antonio Conte has targeted him specifically.
And Chelsea have now responded to the Italian gaffer’s frustration over their lack of transfer activity by bidding £35m for Oxlade-Chamberlain – £10m higher than their original offer.
Meanwhile, Chelsea are reportedly seriously considering a move for Everton contract rebel Ross Barkley.
Ronald Koeman has admitted the England international will leave Goodison Park this summer after refusing to put pen to paper on a new deal.
And The Telegraph claim Conte has his eye on the midfielder after being promised at least three more signings before the transfer window closes.
Tottenham and Manchester United are apparently sniffing around 23-year-old Barkley too.
Finally, Chelsea will not face competition from Tottenham in a potential battle for Joao Cancelo’s signature.
That’s according to The Telegraph, who say Spurs have identified Hoffenheim’s Jeremy Toljan as a more realistic target.
Cancelo, 23, is thought to be on Conte’s radar as he goes in search of a new right wing-back to vie for a starting spot with Victor Moses.
Chelsea like the look of Paris St-Germain and Ivory Coast defender Serge Aurier too.