Press Association Sport understands Walcott has turned down a deal worth around ?4million a year and could be sold before the transfer window closes on Friday.
Walcott reportedly wants to continue negotiations over what is said to be a ?75,000-a-week, five-year contract tabled to replace the one that expires next summer.
It is understood the 23-year-old wants a better deal - nearer ?100,000-a-week according to reports - and should he and Arsenal fail to reach an agreement in the next 48 hours, it could signal the end of his six-and-a-half-year stay in north London.
Walcott has been valued between ?12million and ?15million and has already been linked with Liverpool this summer.
Reports suggest Barclays Premier League champions Manchester City are also interested in adding the England winger to their ever-increasing ex-Gunners contingent after seeing a bid for Swansea's Scott Sinclair rejected.
Walcott's loss would be viewed by many as another huge blow to Arsenal, who lost captain Robin van Persie to Manchester United earlier this month.
Alex Song quickly followed by joining Barcelona, who signed Cesc Fabregas from the Gunners last 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.
Walcott, who joined the Gunners from Southampton as a 16-year-old in 2006, started their home game with Sunderland on the opening day of the season but was left on the bench for Sunday's match at Stoke, eventually coming on after 72 minutes in what was their second straight goalless draw.
Source: PA
Source: PA