With eight months left on his existing deal, the forward can freely talk to overseas clubs from January 1 about a lucrative free transfer next season.
Walcott wants to stay at the Emirates Stadium but the Gunners will have to improve on the £75,000-per-week offer that was turned down in August.
"We are always in touch (with Walcott’s representatives),’ said Wenger.
"He has character, Theo. He keeps focused and he wants to show he is 100 per cent professional and I have a big respect for that attitude."