The Daily Mail says Arsenal want Walcott's long-term future secured within the next fortnight.
Club officials are working to set up a second round of formal contract discussions with the player and his representatives in the next 14 days.
Arsenal are increasingly optimistic a deal can be reached. But the original £75,000-per-week offer made on August 28 must be improved, while Walcott also wants assurances over his chances of playing as a centre forward on a regular basis.
The England star wants a package closer to £100,000 per week, but it is believed a deal of £85,000 or £90,000 would suit both parties.