Wenger has been trailing the Monaco star all summer but the Gunners have failed to get a deal done.
And now United have entered the race for Lemar, with a £55m fee being mooted for the 21-year-old French star.
The Transfer Window podcast guest and former Fleet Street journalist Ian McGarry says Mourinho would love nothing better than getting one over on his managerial rival.
He explained: “Lemar is available, and has been the subject of intense interest from Arenal all the way back to May.
“And so not only would it be a coup for Manchester United, because they would fill that last hole, that last piece of the jigsaw, and finish Jose Mourinho’s business, it would also be a slap in the face to Arsene Wenger.
“And we know Jose Mourinho is one to quite enjoy that.”
McGarry added: “I think this is a very sensible and practical move by Manchester United.
“The Gareth Bale situation around the Super Cup was a bit of a drama, a sideshow, and not neccassarily a very truthful move from Jose Mourinho about his intention of signing Bale.
“It was always my understanding he wanted a more versatile player, someone who could play on both flanks, someone who could drop back, someone who works hard off the ball.”
