The winger's current deal is due to expire in June meaning that, as things stand, he will be free to hold talks with clubs outside of England from January onwards.
Nevertheless, Wenger remains confident of extending the player's stay at the Emirates Stadium.
"I've not thought about losing him because at the moment I think we will still manage to make a deal with him," said Wenger.
"There is urgency [about sorting it out]. We want to sort it out before Christmas, one way or the other."
Walcott, 23, scored his eighth goal of the season last night, as the Gunners were held to a 2-2 draw away at Schalke 04.