Le 10 Sport claims Arsenal had a €14m bid for the United target rejected earlier this month by Griezmann's club, Real Sociedad.
The Spanish side are believed to value Griezmann at €24m - double Arsenal's first offer. But that hasn't put the Gunners off, as the report suggests Arsene Wenger will return with another bid later this month.
But Le 10 Sport also warns that the Frenchman would be more open to a move at the end of the season.
He wants to help Real Sociedad qualify for a European place this term, and with an outside shot of making the French World Cup squad, he reportedly doesn't want to risk a mid-season move.
Having said that, Griezmann told RMC radio recently: "I am relaxed about my future at the moment.
"I have not had talks with other clubs yet. This is something to look at. I want to play for a leading European team and I hope this can help get me a place in the France team."
As well as United and Arsenal, Real Madrid and Monaco have also been linked with the French-born midfielder.
Griezmann primarily plays out wide on the left for Real Sociedad and featured against United in the Champions League group stage this season.