Sky Sports News understands the Brazil midfielder wants to leave Stamford Bridge this summer if Antonio Conte remains as head coach.
Willian has two years remaining on his contract and, at this stage, there has been no formal offer from United.
The 29-year-old, who was voted the club’s players’ player of the season, was left out of the starting line-up for the 1-0 FA Cup final win over United – the fourth time he has been on the bench in the last five games.
He subsequently hinted at a rift between himself and Conte by editing the Italian out of a picture of their trophy celebrations on Instagram on Sunday.
Willian has scored 44 goals in 236 appearances for Chelsea since joining the club from Russian side Anzhi in 2013.