It is understood the 23-year-old gave verbal assurances over his short-term future but has yet to sign a new deal.
Arsenal’s latest offer of a five-year £75,000-a-week contract was rejected by Walcott last Friday but the club are now convinced the player wants to stay.
Manchester City remain interested and could yet test Arsenal’s resolve.
There is, however, a feeling that a breakthrough has been made with senior figures at Arsenal now resolved to allow Walcott to enter the final 12 months of his contract, despite the risk of losing him for nothing next summer.
The change of mood came after talks between player and manager at Arsenal’s London Colney training ground, where Walcott left Wenger in no doubt about his commitment to the club.
Arsenal had previously been prepared to sell their winger if he did not agree to sign a new contract by Friday, however that position has now softened.
Further talks are planned in the weeks ahead, with both parties having made a commitment to work towards a resolution.
Walcott, who played for England in Euro 2012, joined Arsenal from Southampton for £12.5m as a 16-year-old in 2006.
He has scored 42 goals in 222 appearances and won 28 England caps during his time at the club.
Last season, he contributed 12 assists in the Premier League, many of which were for Robin van Persie.
Striker Van Persie moved to Manchester United for £24m on 17 August, three days before Alex Song joined Barcelona.
Walcott started the home draw with Sunderland on the opening day of the season but was left on the bench for Sunday’s draw at Stoke.