The Daily Mail says Arsenal are increasingly confident of persuading Walcott to sign a new five-year deal.
Talks with Walcott's representatives over a new contract are ongoing, and the club hope to finalise a deal within a fortnight.
There is now a growing sense at the Emirates that they are winning the battle to secure his future.
A number of the forward's team-mates are convinced he will stay, as are members of the club hierarchy.
Walcott has indicated that he wants to stay but feels Arsenal are short-changing him with their current £75,000-per-week offer.
They are expected to improve that this month but the increase will decide whether Walcott signs an extension. A deal between £85,000 and £90,000 would satisfy both parties.