The Sunday Mirror says Wenger has offered £5 million for the Moroccan striker - a bid that was dismissed by Bordeaux as well short of their valuation.
Their manager Laurent Blanc values the 25-year-old at twice that amount.
But Wenger is aware that Chamakh is out of contract at the end of the season and will be able to get him on a free transfer under the Bosman ruling.
He can also start talking to Chamakh in January to negotiate a deal for next season - and Blanc would then have a distracted player on his hands.
Wenger wants to add to his squad before the deadline and is now set to go in with a final offer close to £8.5m in the hope that Bordeaux will relent and allow the player to leave.