According to Corriere dello Sport (h/t the Mirror), Walcott is hoping to leave the Emirates Stadium in search of more game time ahead of the FIFA World Cup, and he believes the Rossoneri could give him the minutes he desires.
Milan are said to be looking for a central midfielder to provide cover for Franck Kessie rather than a forward but could be open to taking the 28-year-old on loan for the remainder of the season.
Walcott has played just 49 minutes of Premier League football for Arsenal this term, becoming something of a forgotten man, as the Press Association's Mark Mann-Bryans noted:
Mark Mann-Bryans @MarkyMBryans
Walcott on the sidelines with a groin injury, Wenger reveals he is not in the squad to face Newcastle. Quite damning that he wasn't even mentioned in earlier team news question... #AFC2017-12-15 08:58:56
That he has played so little is perhaps a surprise given he netted 19 goals in all competitions last season, his second-most prolific campaign with the Gunners during his 12 years at the club.
The England international has made almost 400 appearances for Arsenal since he joined as a 16-year-old, but his career has been punctuated by the feeling he has not reached his potential, though his inclusion in the Three Lions' 2006 World Cup squad will have done him no favours in contributing to the lofty expectations placed on him.
One aspect of Walcott's game that often lets him down is his delivery. Despite spending most of his career on the right flank, his final ball still leaves much to be desired, per ESPN FC's Mattias Karen:
Mattias Karén @MattiasKaren
Among the things that never fail to disappoint with Arsenal: Walcott's crossing ability... #ARSWHU2017-12-19 20:07:28
As his goal tally last year showed, though, he still has something to offer Arsenal's squad—or Milan's.
He hasn't lost his pace, and there are times when the Gunners are in need of that, as Football.London's Charles Watts noted:
Charles Watts @charles_watts
I would 100 per cent get Walcott on for the second half. Crying out for a bit of pace, someone to go beyond Lacazette. So static from Arsenal right now, easy to defend against.2017-12-31 17:24:38
It could be time for him to leave Arsenal. A change is needed if he's to have a chance of representing the England in Russia this summer.
A switch to Serie A is among the more interesting moves he could make. His speed could allow him to stand out amid the slightly slower pace of Italian football, so it may be a good option for him if Milan are willing to take him on.