The Chilean is expected to leave Arsenal in the summer, although the Gunners could allow Alexis to leave the club during the January transfer window if they are able to recoup most of the £38m they paid to Barcelona in 2014. The Toffees are looking to strengthen their attacking options, despite the recent signing of Turkish worldwide Cenk Tosun from Besiktas.
Chelsea defender Andreas Christensen, 21, has signed a new four-and-a-half-year contract.
Walcott has struggled to secure regular playing time at Arsenal this season.
Walcott has been heavily linked with a move away from the Emirates Stadium this month having failed to make a single Premier League start all season, although manager Arsene Wenger recently insisted that he wants the 28-year-old to stay at the club.
Walcott came through Southampton's youth system and, at just 16, became the club's youngest ever player to appear for the senior team.
Walcott's directness would add a lot to Southampton's game, while he managed to score 19 goals in 37 games across all competitions last season.