The 23-year-old has 12 months to run on his existing deal at the Emirates and is yet to hold discussions over an extension.
Jurgen Klopp is a huge fan of the midfielder and the Reds have already held initial talks with Oxlade-Chamberlain’s representatives.
Liverpool hope to lure Oxlade-Chamberlain to the club by promising him a central midfield role and the Daily Express claim they are willing to pay up to £40m to sign him.
That would eclipse the £35m the Merseyside giants paid for Andy Carroll in 2011 but they need to convince Oxlade-Chamberlain to make the move first.
The Gunners remain hopeful of agreeing on a new contract with the midfielder but Oxlade-Chamberlain has reservations over his position in the side.
Should he show no signs of signing a new deal, Arsenal will look to prise as big a fee as possible for Oxlade-Chamberlain from Liverpool.
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