Walcott, who is yet to start a Premier League game this season, has talked to Gunners manager Arsene Wenger about his lack of playing time, per the report.
However, O'Rourke also noted how Wenger may opt to keep the forward amid doubts about the futures of Alexis Sanchez and Mesut Ozil, who are out of contract in the summer.
Their situations aren't the only concern for Wenger, with striker Olivier Giroud injured. The Arsenal boss has ruled the France international out until mid-January with a hamstring problem, per John Cross of the Daily Mirror.
Arsenal may be short in attack, but it's notable Walcott still hasn't been getting starts in the league. He was on the bench for the 1-1 draw against West Bromwich Albion on New Year's Eve, but he was still overlooked in favour of Danny Welbeck when Wenger opted for substitutions.
Wenger may have been looking past Walcott recently, but the 28-year-old still has value.
He has scored three times in helping Arsenal qualify for the last 32 of the UEFA Europa League. The England international also scored the winner to beat Doncaster Rovers in the third round of the Carabao Cup, a competition in which the Gunners have reached the last four.
The two-legged semi-final against London rivals Chelsea later this month will test the strength in depth of Wenger's squad. So will the FA Cup third-round tie against Championship side Nottingham Forest on Sunday. Arsenal are the holders and are aiming for a fourth win in the last five seasons.
Wenger still needs Walcott, but if the forward pushes for a move, a return to boyhood club Southampton could suit him. He moved from Saints to the Gunners as a 16-year-old in 2006.
Alternatively, Everton manager Sam Allardyce has made little secret of his desire to add a striker, per David Maddock of the Daily Mirror: "We need to strengthen the front line. We're trying to do the hardest job of all by securing a player in that position."
Walcott is unlikely to start ahead of Sanchez and Alexandre Lacazette, but an Arsenal squad in a position to build on recent cup successes will still find his pace and eye for goal useful.