Walcott has slipped further down the pecking order at the Emirates Stadium this season and has yet to start a Premier League game, while he has started on the bench on just three occasions.
The 28-year-old is desperate to play regular first-team football once again, and Allardyce, who will be handed a large transfer kitty in the January transfer window, is ready to offer him that chance.
Allardyce has impressed since taking charge at Everton and after taking the Toffees to ninth in the Premier League, will be handed the funds to help move the club on to the next level.
Walcott still has 18 months remaining on his current contract with the Gunners but it is clear he no longer figures in the plans of Gunners boss Arsene Wenger and will be allowed to leave if the price is right.
The forward will not be short of offers if he is allowed to leave, but Allardyce will be confident of fighting off the competition to strike a deal.