The 24-year-old turned down a move to Chelsea - worth in the region of £35-40 million - because he favours a switch to Merseyside and is holding out for a late bid from Liverpool.
The Gunners were willing to sell Oxlade-Chamberlain to the Premier League champions, but are reportedly opposed to selling him to to Liverpool because they see them as closer rivals.
The England international is keen to play in his preferred position of central midfield and believes he has a greater chance of doing so at Liverpool, with Chelsea having offered him the chance to take up a wing-back role.
Arsenal would not be against keeping Oxlade-Chamberlain - in spite of the player's desire to leave - if they cannot come to an agreement with Liverpool.
Oxlade-Chamberlain, who can play out wide as well as centrally, has one year remaining on his contract at the Emirates Stadium. He will leave for free next summer if the Gunners don't sell him before the deadline shuts on Thursday night.
Reds deciding between Lemar and Oxlade-Chamberlain?
But Liverpool must decide whether to chase a deal for Oxlade-Chamberlain, who they view as an alternative to Monaco midfielder Thomas Lemar.
The Reds submitted a £75 million offer for the Frenchman with the Ligue 1 champions yet to respond.
Deals will be difficult to conclude for either player, with Liverpool unlikely to bid as high as Chelsea given Oxlade-Chamberlain is out of contract in North London in a year.
Oxlade-Chamberlain, who has made 198 appearances at Arsenal and scored 20 goals since joining from Southampton in 2011, has started all of Arsenal's games so far this season.
He lasted 62 minutes of their 4-0 thrashing at the hands of Jürgen Klopp's charges on Sunday at Anfield, meeting Arsène Wenger beforehand to reiterate his desire to leave.