According to Sami Mokbel in the Daily Mail, the Toffees have joined Saints in their pursuit of Walcott, who is valued at £20 million, and feel confident they'll come out on top as they're better equipped to match or at least come close to his £110,000-per-week wage.
By contrast, he'd have to take a pay cut if he were to sign for Southampton, the club he left when he joined Arsenal as a 16-year-old.
Walcott has played very little this season, with his total of 49 minutes of Premier League football coming across five appearances.
Football writer John Brewin believes the forward should have moved on from the Gunners by now:
The 28-year-old was handed 90 minutes in the FA Cup on Sunday, but he failed to impress as Arsenal were beaten 4-2 by Nottingham Forest.
ESPN FC's Mattias Karen suggested Walcott may not have another opportunity like that again in north London:
While he has over 100 goals for the team, the England international has never quite met expectations at the Emirates Stadium, and his departure would be unlikely to upset many supporters.
The forward's pace is a big asset, but he lacks the technical ability and end product to excel at a club such as Arsenal.
That's not to say he couldn't be a useful acquisition for Everton or Southampton, though.
Walcott scored 19 goals in all competitions last year, and this season he has produced three goals and five assists in five UEFA Europa League appearances.
The Toffees have scored 25 goals in 22 Premier League matches this season, while Southampton have scored just 21, so there's need for more firepower at both, and Walcott could help in that regard.
Both clubs would also be able to offer him more regular football ahead of the World Cup, so a move could benefit all parties.