The Gunners came close to landing the French winger with an incredible £92million bid on deadline day.
Mourinho was frustrated at the club's failure to land a winger in the summer and after failing to land Ivan Perisic, he's turned his attentions to Lemar.
The player's representatives are reported to be aware of United's interest and he is believed to be open to a switch to Old Trafford.More news: 3 defensive keys for Cardinals in Week 1 vs. Lions
More news: Liverpool manager Jurgen Klopp: All good with Philippe Coutinho
More news: Aaron Donald reports to the Rams, ending his holdout
Monaco, who won Ligue 1 last season and managed to make the semi-finals of the Champions League, sold off multiple key players during the transfer window, including Kylian Mbappe and Benjamin Mendy.
Lemar's brimming potential, youthful exuberance and versatility to play in any attacking position is undoubtedly something that Arsenal could benefit from, especially after Alex Oxlade-Chamberlain's departure to Liverpool.
Any move for the France worldwide will likely top the £89m club-record fee shelled out for Paul Pogba past year.
While Arsene Wenger is set on continuing his pursuit of Lemar in January as the replacement for Alexis Sanchez, Jose Mourinho has plans to intercept Lemar's signature.
The Chile global is expected to complete a move to Manchester City in January for a knockdown price of £20m.