The news was relayed by Paul Joyce of The Times on Tuesday:
BBC Sport's David Ornstein provided more details, including where the money recouped by the Gunners may be spent:
David Ornstein @bbcsport_david
Everton close to agreement with Arsenal to sign Theo Walcott for in excess of £20m. #EFC football/financial offer beat rivals. Likely played last #AFC game (397 apps, 108 goals). Money to go towards transfers - Aubameyang/Malcom + Mkhitaryan (Sanchez swap), unlikely to get all 3
According to Bleacher Report's Dean Jones, a £20 million fee has been agreed between the two sides:
Mark Critchley of The Independent later reported Walcott has undergone a medical—which Sky Sports claim he passed before agreeing personal terms:
Walcott has fallen out of favour at the Emirates Stadium and has yet to start a Premier League game this season for the Gunners.
As noted by Andy Hunter of the Guardian, Everton manager Sam Allardyce confirmed his interest in the player last week, and the Toffees are said to be confident of getting a deal completed for the England international this week. Southampton, where he started his senior career, were also noted as an interested party.
Everton have been lifeless in attack in recent weeks, and in their 4-0 loss to Tottenham Hotspur on Saturday, they failed to offer anything in the final third.
These numbers from Sky Sports' Matt Cheetham summed up just how poor Everton have been in an attacking sense as of late:
Walcott seems like a sensible option with that in mind, as the forward can bring goals, creativity and searing pace to any side when at his best.
However, little of that this season has been seen, as Arsenal manager Arsene Wenger has opted to use other attackers ahead of the 28-year-old. Alexandre Lacazette, Olivier Giroud, Alexis Sanchez, Alex Iwobi and Danny Welbeck have all pushed ahead of Walcott.
As a result, a move is necessary for Walcott in January, especially if he is to ignite any slender hopes of forcing his way into contention for England's squad for the 2018 FIFA World Cup.
Should the transfer go through, Walcott will at least be getting regular minutes and would be a vital part of Allardyce's Everton attack.
Per football writer Paul Brown, while the forward has struggled this season, last term he showed he can still put the ball in the back of the net:
There are reasons why Walcott has fallen down the pecking order at Arsenal, though, with his fluctuating form being the main one.
While his pace always poses a threat to opposing defences, he's yet to fully nail down a regular position in his career, moving between the flank and the centre-forward position during his time at Arsenal.
There are areas of his game where he falls short. Walcott can make poor decisions when trying to knit moves together, and his final ball can leave something to be desired. For all of Everton's woes, journalist Joe Strange isn't convinced it's the right move:
Walcott has produced some moments to remember in his 12 years at the club. However, the player, his manager and the supporters will all accept it's time for a change of scenery.
The forward will look at how a move to Liverpool has recharged former Arsenal team-mate Alex Oxlade-Chamberlain and will be hopeful a switch to Merseyside can do the same for his fortunes. At Goodison Park, he will at least get a chance to remind the football world what he's capable of.