Press Association Sport understands Walcott did not back down over his contract demands but gave assurances he definitely wanted to stay with the Gunners in the long-term.
Arsenal had been braced for the prospect of selling Walcott before the transfer window closes on Friday but they are now happy to continue negotiations beyond then and effectively risk losing him for nothing next summer.
That is a scenario the club have not been prepared to tolerate in the past, having decided to sell a host of big name players in recent years as they have entered the final year of their contract.
The difference with Walcott is that there is genuine hope an agreement can be reached, although how Arsenal and manager Wenger would react to a big-money bid in the next 48 hours remains to be seen.
Walcott, who has been valued between ?12million and ?15million, has already been linked with Liverpool this summer.
Reports suggest Barclays Premier League champions Manchester City are also interested in adding the England star to their ever-increasing arsenal of former Gunners after seeing a bid for Swansea's Scott Sinclair rejected.
It is understood Arsenal offered Walcott a contract worth around ?75,000-a-week, reportedly for the next five years.
But the player himself is after a better deal - nearer ?100,000-a-week according to reports, while he is also seeking assurances about his position in the team.
Losing Walcott would be viewed by many as another huge blow to Arsenal, who sold captain Robin van Persie to Manchester United earlier this month.
Source: PA
Source: PA