The attacker has made only seven starts for the Gunners this term, with them all arriving in either the Europa League or EFL Cup.
The limited game time means Walcott is now almost certain to make Gareth Southgate’s England squad for the World Cup in Russia.
The 28-year-old has come off the bench three times in the Premier League but he has struggled to make an impact.
Manager Arsene Wenger is expected to lose Mesut Ozil and Alexis Sanchez in January with the pair out of contract at the end of the season.
The club are thought to be trying to sell their star assets instead of risking them walking away for free.
Jack Wilshere is also considering a move away from the Emirates Stadium to try and win a place on the plane to Russia.
But Walcott wants to try and force his way into Wenger’s plans once the aforementioned trio depart.
But if he is unable to do so then El Gol Digital report he will be offered an olive branch by Valencia.
It is also suggested Valencia will try to tempt Atletico Madrid forward Kevin Gameiro to the Mestalla Stadium.
The La Liga outfit are currently second in the table and are leaders Barcelona’s closest challenger.
However, they still trail Ernesto Valverde’s outfit by five points.