The Sunday Mirror claims Red Devils boss Jose Mourinho is an admirer of the playmaker and could be persuaded to launch a move for a player he coached at Real Madrid.
Ozil, 28, has reportedly decided to reject signing a new Arsenal contract, which means he will be free to discuss a Bosman move with foreign clubs from January onwards.
However, the Germany international is understood to want to stay in the Premier League and could potentially join one of Arsenal's rivals in a free transfer next summer.
The Mirror claims Ozil's representatives have been encouraged by the feedback they have received from United to their proposal for a transfer.
Arsenal boss Arsene Wenger admitted earlier this month that contract talks with Ozil had "slowed down" during the early weeks of the season.
"The contract talks have slowed down a little bit since August 31, because we need a little breather and to focus on the competition," he was quoted as saying by the Independent on 13 September. "But we will slowly get back into that."
There is also widespread speculation over the future of Ozil's teammate Alexis Sanchez, who is in the final year of his contract at the Emirates Stadium.
The Chilean forward tried to force through a move to Manchester City during the summer transfer window but failed, while French giants Paris Saint-Germain are also reported to be interested in his services.
Ozil has started four league games for Arsenal this season, but missed the Gunners' trip to Chelsea last weekend due to a knee problem. He also sat out the midweek Carabao Cup win over Doncaster.
The German playmaker has returned to training and is in contention to start against West Brom at the Emirates Stadium on 25 September.