Arsenal have tried to tie Oxlade-Chamberlain down to a new contract, but so far, the player's representative has rejected all the offers.
The Mail claim that while Arsenal have rebuffed "initial overtures", Chelsea are set to "intensify" efforts.
Arsenal are keen to keep hold of the 23-year-old, but with just one year remaining on his current deal, and the Gunners being reluctant to meet his wage demands, they could be forced to cash in or face losing him for nothing next summer.
The Daily Star note in their report that Oxlade-Chamberlain would welcome a move to Stamford Bridge, which comes as no surprise - who wouldn't?
Liverpool and Chelsea have both shown an interest in the player this summer.
Arsene Wenger has made it clear to Arsenal's board that he does not want to lose Oxlade-Chamberlain to another Premier League rival.
Speaking ahead of Arsenal's first game of the new season against Leicester City, Wenger said of Oxlade-Chamberlain: 'I think in his career he had some bad moments because he had some small injuries.
"Last year he was much more stable". Personally I believe he is a hugely talented player who is on the way up.
Oxlade-Chamberlain has been linked with a potential move to Chelsea and Liverpool in a £25m move but a transfer deal has failed to materlialise so far. Today he is conscious of his qualities.