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 it is now happy to continue negotiations beyond then and effectively risk losing him for nothing next northern summer.
That is a scenario the club has 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 £12 million ($18.36 million) and £15 million ($22.95 million), has already been linked with Liverpool this northern summer.
Reports suggest Premier League champion Manchester City is also interested in adding the England star to its ever-increasing arsenal of former Gunners after seeing a bid for Swansea City's Scott Sinclair rejected.
It is understood Arsenal offered Walcott a contract worth around £75,000 ($114,739)-a-week, reportedly for the next five years.
But the player himself is after a better deal - nearer £100,000 ($152,986)-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, which sold captain Robin van Persie to Manchester United earlier this month.
Alex Song quickly followed by joining Barcelona, which signed Cesc Fabregas from the Gunners last northern summer, which also saw Gael Clichy and Samir Nasri move to City.
Walcott was one of Van Persie's main providers last season as the Dutchman notched 37 goals, with Walcott himself scoring 11.
He has often been criticised for a lack of consistency at both club and international level and Arsenal team-mate Alex Oxlade-Chamberlain was preferred by England manager Roy Hodgson at EURO 2012.
Walcott, who joined the Gunners from Southampton as a 16-year-old in 2006, started its home game with Sunderland on the opening day of the season but was left on the bench for last weekend's match at Stoke City, eventually coming on after 72 minutes in what was its second straight goalless draw.
A lack of goals has seen Wenger's northern summer signings come under scrutiny but one of them, Olivier Giroud, has denied he is feeling the pressure of helping fill the shoes of Van Persie.
"I'm not the kind of person to put negative pressure on my shoulders," the France striker told Arsenal Player.
"It's true that Robin had a very, very good season last year but I'm not here to make everyone forget about him or to replace him.
"I only want to take my place, gradually make my mark and win over the hearts of the fans.
"In our way, we are two different players. He spent eight years here and had a very good season last year.
"I need a little adjustment time but I hope to make my mark as quickly as possible. And I hope to score a lot of goals."
Giroud, who signed for Arsenal after a prolific campaign with Montpellier, claimed playing in front of fellow new boy Santi Cazorla would help him adapt.
"It's the same gameplan that we had at Montpellier and the same plan I played with in the French national team, with a number 10 like Cazorla behind me," he said.
"It's the same tactical plan I am used to so I am not lost, I know how to play in relation to my partners and it's just a little adjustment that is needed to make things automatic on the pitch."
English Premier League
Newcastle United boss Alan Pardew was keeping his own counsel on Tuesday after the club finally confirmed Joe Kinnear's arrival as director of football.