The report claims that both parties are aiming to thrash out the final details of the deal in time for Walcott to have the relevant medical checks.
Walcott, a 28-year-old English worldwide, has been at Arsenal since joining the club from Southampton in 2006.
Despite a record of 108 goals in nearly 400 appearances, however, Walcott never truly lived up to his potential at the Emirates, though he was hampered by injuries at times too.
Should Walcott come through his medical without any problems, he will become Everton's second signing of the month following the arrival of Turkey worldwide Cenk Tosun from Besiktas. Likely played last #AFC game (397 apps, 108 goals).
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Everton's offer is expected to fend off rivals, including Southampton.
Walcott, Arsenal's longest-serving player, has yet to start a Premier League game this season and Arsene Wenger has sanctioned a move away for the winger.
Talks are believed to have been held between the two Premier League sides on Monday amid speculation that Everton will stump up £20m for Walcott, and Arsenal are only too happy to sell a player who is now surplus to requirements.
The Toffees have struggled in sustaining attacks this season and Walcott's pace and goalscoring record have caught Sam Allardyce's eye as he looks to support new signing Cenk Tosun up front.More news: Angelique beats Ashleigh to win Sydney International