The Brazil international has been linked with a potential move to the Red Devils after he fell out with Chelsea FC boss Antonio Conte.
Willian has already been the subject of a £50m bid from the La Liga giants with view to a switch to Camp Nou ahead of the new season.
However, Chelsea FC are reported to have turned down the offer from the Spanish champions as the Blues look to secure maximum value.
Manchester United are no strangers to signing top players from their direct title rivals under Portuguese head coach Jose Mourinho.
The Red Devils signed Serbia international Nemanja Matic in a £40m deal from Chelsea FC last summer to bolster their midfield options.
Manchester United completed the signing of Alexis Sanchez on a free transfer from Arsenal in the January transfer window.
Sky Sports News reporter Downie believes Willian is a great position as he looks to secure the final big contract of his career.
“What a position he is in,” Downie told the Transfer Talk podcast. “He is 29 and the last big contract of his career.
“He has two of the biggest clubs in the world chasing him – Barcelona and Man United. He is a typical Mourinho signing.
“He has had him before. He knows his mentality. He is very keen on him. When Barcelona come calling, it is hard to say now.”
Willian has won two Premier League titles, the League Cup and the FA Cup during his five-season stint at Stamford Bridge.
The Blues haven’t signed any players in the summer transfer window so far despite finishing in fifth spot last term.
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