Allardyce is looking for some quality reinforcements to a strike force that currently consists of Wayne Rooney, Dominic Calvert-Lewin and Sandro Ramirez.
Walcott may be available in a cut-price deal as his contract expires at the end of next season and given his current amount of playing time he may refuse an extension.
The 28-year-old can also provide an option out on the wings, a position he has played at frequently for Arsenal and a position in which Everton lack players.
The Arsenal man was included in Allardyce's solitary England squad in the 63-year-old's short stint at the helm of the national side, coming on as the second substitute in England's 1-0 win over Slovakia.
Struggling to replace Lukaku
Despite spending almost £150m this Summer, Everton failed to replace Romelu Lukaku following the Belgian's transfer to Manchester United, a decision that was roundly criticised amongst football professionals but the new Toffees manager defended the decision earlier this year.
Speaking to Paddy Power before joining Everton, Allardyce said he "understands why they didn't replace Lukaku" because "nowadays that's such a hugely difficult piece of business" and there is "such a lack of top quality strikers."
“But still, the club needs someone to get goals and change those results around.”
On the other hand, Sandro, who was brought in alongside Wayne Rooney to try and plug the gap left by Lukaku, could be on his way out after struggling to settle in at Goodison Park.
Benchwarming at Arsenal
Walcott has struggled to break into the Arsenal line-up this season after the arrival of Alexandre Lacazette, making only three Premier League appearances this campaign.
Though his time at Arsenal has been plagued with long spells on the sidelines through injury, Walcott has scored 107 goals in 391 appearances during his 11 years with the Gunners.
Last season was one of the Englishman's best scoring 19 goals in 37 appearances in all competitions despite Arsenal's disappointing season, failing to qualify for the Champions League.
Allardyce could be hoping Walcott can continue this form and lead the line at Goodison Park as he tries to drag them up the table.