The 28-year-old has fallen out of favour with the 'Gunners' this season, featuring in just three Premier League games.
However, he has shown his worth in the Europe League, having registered three goals and five assists in just five games in Europe.
Walcott has just 18 months remaining on his £140,000 a week deal at the Emirates and is thought to be keen on a move in January as he looks to seal regular football ahead of next summer's World Cup in Russia.
Everton boss Sam Allardyce is a keen admirer of the former Southampton man, with 'Big Sam' keen on adding more natural width and pace to the side he inherited from Ronald Koeman.
Allardyce has impressed early on at Goodison Park and is likely to be given significant funds to strengthen in January, with Walcott and Sevilla's Steven N'Zonzi thought to be his top two targets.
The signing of Walcott would be welcomed by certain sections of the Everton faithful, with many fans having grown tired of seing players forced to play out of position to plug gaps out wide, commenting that it is a case of 'square pegs in round holes'.