Oxlade-Chamberlain has made it clear to Arsenal he wants to leave before the transfer window closes on Thursday.
Liverpool and Chelsea both interested in signing the versatile midfielder.
But Express Sport understands Chelsea have stolen a march Liverpool by agreeing a £35m fee with Arsenal for his signature.
The 24-year-old is said to prefer a switch to Anfield but Antonio Conte is keen to bolster his midfield options with Oxlade-Chamberlain able to occupy the wing-back positions.
But the Irish Independent claim Liverpool could return with a bid of their own should they fail to sign Thomas Lemar from Monaco.
The Reds have already had two bids rejected for the France international.
Monaco have repeatedly insisted Lemar is not for sale all summer with Aesene Wenger confirmed last week Arsenal had ended their interest in the left-sided winger.
However, Liverpool have stepped in to test Monaco’s resolve and are expected to return with a third bid.
Should they be knocked back once more, it is said the Reds will switch their attention back to Oxlade-Chamberlain.
Liverpool kicked off the week with a bang after sealing a club record deal of £48million to sign Naby Keita, who will move to Anfield next summer.