Wenger, 68, made the surprise announcement last week that he will end his 21-year reign at Arsenal when the season is completed.
His rivalry with Mourinho often spilled over into physical altercations dating as far back as when the Portuguese took over at Chelsea in 2004 and promptly grabbed Premier League supremacy from Wenger’s Arsenal team.
Wenger had not got the better of Mourinho in a competitive fixture until Arsenal beat United 2-0 at the end of last season.
Mourinho has greeted Wenger’s announcement warmly and even suggested they could “even be friends in the future”.
“It depends,” said Wenger. “I respect him of course… you should give me a little bit of peace for my final weeks and not try to push me into another confrontation.
“I want to go peacefully, and with Mourinho now because he is a great manager. I say goodbye to everyone. In France we say ‘au revoir’ which means you might see me again.”
Before Mourinho, Wenger frequently clashed with Alex Ferguson in Arsenal-United matches, but former United chairman Martin Edwards revealed in 2017 that he held talks with Wenger about replacing Ferguson when the Scot originally announced he would retire in 2002.
Ferguson changed his mind and would stay for another 11 years while Wenger said at the time Arsenal’s “values” meant he would never have accepted the job anyway.
“I turned many clubs down to stay here and face the challenge when we built the stadium. I don’t regret that because I see my life and what is important to me,” he said on Friday.
The Gunners go to Old Trafford ahead of the second leg of their Europa League semi-final against Atletico Madrid, with the tie delicately balanced after a 1-1 draw at the Emirates on Thursday.