It is claimed United are willing to offer the Frenchman a £216,000-a-week contract to tempt him into a move to the Premier League.
Mourinho is also hoping Paul Pogba will help in convincing Greizmann, who is said to have a £85.5million release clause in his contract, into a transfer to Manchester.
Paul Merson reckons Arsenal should hijack Man United's move for Antoine Griezmann
But former Arsenal midfielder Merson has urged the Gunners to battle with their rivals to sign Griezmann and add quality to the side rather than fret about possibly losing Alexis Sanchez and Mesut Ozil.
“They pay top dollar those Arsenal fans,” Merson told talkSPORT.
“When you’re paying that kind of money you have to be watching world class footballers at the club. You can’t just have two – you should have five and six in my opinion.
Paul Merson reckons Griezmann is a 'world class' player and would suit Arsenal
Jose Mourinho is keen on bringing Griezmann to Old Trafford in the summer
Arsenal's potential January ins and outs
Sun, December 25, 2016
Express Sport takes a look at the star players Arsenal could sign and who could leave in January
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Dimitri Payet: Open to a move to the Emirates saying it would be ‘fun’ to play for Wenger
“With Griezmann, people go ‘they’ve [Arsenal] got that kind of player’. They haven’t. They’ve got two of them but you could do with another one.
“For me there isn’t a club in the world that wouldn’t want Griezmann.”
Griezmann has scored 13 goals in 26 appearances for Atletico this season and is contracted to the club until 2021.