Oxlade-Chamberlain, 23, is about to enter the final 12 months of his Gunners contract.
Arsenal are keen to tie him down to a new deal.
But, according to the Mirror, talks have reached a standstill.
There is apparently a growing feeling at the Emirates that Oxlade-Chamberlain could be on his way out of the club.
And Liverpool are reportedly looking to capitalise on the uncertainty surrounding his situation.
The Mirror claim the Reds have already made contact with Arsenal over a deal to take him to Anfield.
They are apparently hoping to steal a march on Premier League rivals Chelsea and Manchester City, who are said to be watching developments closely.
The Mirror say the Gunners rejected Liverpool’s approach out of hand.
But they suggest the north London outfit could be tempted into selling Oxlade-Chamberlain rather than risk losing him for nothing next summer.
Liverpool have made an approach to sign Alex Oxlade-Chamberlain.
The midfielder, apparently valued at around £25m, has failed to nail down a starting spot with Arsenal since his arrival from Southampton in 2011.
He made 29 Premier League appearances for Arsene Wenger’s side last season but only 16 of those came as a starter.