The summer transfer window has been open for quite a while in England and it closes on 9 August - just before the new season kicks off.
Juillard told followers on social media that Fellaini will meet with Arsenal reps "at the beginning of the week".
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.
The midfielder is set to become a free agent at the end of this month should the Red Devils fail to agree to a new deal with the Belgian in the coming weeks.
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Fellaini, who is part of Belgium's World Cup squad, joined United from Everton in 2013 but has failed to hold down a regular place in recent times. He also wanted a significant pay-rise.
Arsenal have both Jack Wilshere and Santi Cazorla out of contract this summer, with the latter signing for Villareal which means midfield recruitments are necessary going into the 2018/19 season.
The 30-year-old is believed to be in talks with several clubs already, namely AC Milan, Paris Saint-Germain and West Ham United.
However, if they are found guilty, chances are that Fellaini may not move to Italy and would rather stay in England to continue his career.