Sam Allardyce got his Everton tenure off to a winning start with a 2-0 victory over Huddersfield Town and the new manager will be eager to strengthen his squad when the transfer window opens in January.
Walcott has mainly featured in cup competitions for Arsenal this season and his contract expires in 2019, with his future at the club looking increasingly uncertain.
Having not been called up by his country for over a year, a move to Everton, where he would in all likelihood play regularly, would give Walcott a chance of getting into Gareth Southgate's England squad for next summer's World Cup.
Although he was unable to fulfil his potential at Arsenal, Walcott has still had many positive moments during almost 12 years with the club and would be an excellent addition to Everton's squad.
Aaron Lennon has been performing well for Everton of late and the Toffees also have Yannick Bolasie to return from injury, but Walcott is also capable of playing on the left flank at times and he has been used as a striker.