The World Cup winner with Germany's current deal at the Emirates runs down after the end of the 2017/18 season. He has joined Arsenal forward Alexis Sanchez on the players who are expected to leave the club on a free transfer.
Arsene Wenger has already admitted "it would be possible" to cash in on both Ozil and Sanchez on a free transfer in the January transfer window. According to the Manchester Evening News, the player is believed to be keen on making a switch to Old Trafford as a free agent next summer.
Any move from the 20-time English champions could see them face competition from Barcelona. According to the Mirror, the Catalan club are willing to offer Ozil £350,000-a-week ($466,742 per week) to join them for a free transfer next summer.
However, Wright believes the German international will join Jose Mourinho's side and admitted the potential transfer of Ozil to United will break his heart.
"That [Ozil's transfer to United] would break my heart. I could see him at United being unbelievable. Mesut Ozil on a free to Manchester United looks a shoo-in," Wright told BBC Radio 5 Live.
When asked if he would make a difference, Wright said: "Yes, we have seen it sometimes at Arsenal. I've said to people before. When you have players around, he can't drag a team up by their laces and make them play well."
"If they are playing well, you stick him in. You know what, 'I just need someone to finish this off and I put Ozil in.' That frightens me to know he could end up at Man Utd. It wouldn't make sense for him or his agent or his team to sign that [Arsenal deal] now simply because it has gone this far.
"What Arsenal fans will be frightened to see is to Ozil go somewhere like Manchester United and they get the best out of him. Because Mourinho will not take somebody not performing. You can see it, and that frightens me to death."
Ozil has played under Mourinho when the pair were at Real Madrid. He could reunite with the Portuguese tactician at United after this campaign. He has scored one goal and registered three assists in 10 matches in the Premier League this term.