The Portuguese manager, prior to the end of the Premier League season, had mentioned his confidence over Fellaini's future, but the midfielder is set to enter fresh contract talks.
But the report claims that Emery has made contact with the midfielder since he was officially appointed as Arsene Wenger's successor less than two weeks ago. According to Juillard, Unai Emery's Arsenal are the first to pounce, otherwise Fellaini may well become a free transfer this summer.
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Paul Hirst of The Times reported on Monday afternoon that Arsenal's interest in Fellaini is genuine and he is more likely to move to the Emirates Stadium than put pen to paper on a new deal at United.
Fellaini, who made his name at Everton, is now away with Belgium's World Cup squad and is not expected to make any decisions over his future until he returns from Russian Federation next month.
The Belgian is available on a free transfer at the end of his month when his United contract expires, and there has been little evidence thus far that the Old Trafford club will look to offer Fellaini a new deal.
Fellaini has also been linked to Paris Saint-Germain and Marseille.