Wenger upset Stoke last month with his comments regarding the team's style of play and he also singled out defender Ryan Shawcross for criticism.
The Potters complained to the Premier League and the Football Association in response to the comments, but no action was taken against Wenger.
However, chief executive Tony Scholes says Stoke will not allow the Gunners boss to maintain his criticism of the club and will continue to pursue the issue with the footballing authorities.
"Many of you will have seen the comments made by Arsene Wenger in which he was critical of the club generally and Ryan Shawcross in particular," said Scholes.
"We have written to him objecting to these comments and have asked for an apology.
"Much as we respect Arsene Wenger, we cannot allow him to continue criticising us in this way.
"Therefore, in the absence of any apology, we will continue with our complaint, even though it has been reported that the FA will take no action.
"We cannot accept that. It is not right that anyone in the game can criticise one of our players with reproach."