The Brazil international's first team appearances have been restricted since he lost his place in the starting lineup to Pedro. He has made 13 starts and made 17 appearances from the bench in the Premier League.
According to Goal.com, Willian has grown frustrated with the limited role at Chelsea. The Red Devils will be hoping to use his lack of playing time at Stamford Bridge as the reason to leave the Blues for United.
The South American winger was regular under Antonio Conte earlier in the season. However, he took time away from the first team following his mother's death in October. The situation allowed Pedro to impress the Italian manager that has seen him cement his place in the starting lineup.
He has played under Jose Mourinho, when the Portuguese tactician was in charge of the west London club. United are hoping that they can help the winger reunite with his former manager at Old Trafford.
Chelsea parted ways with Mourinho in December 2015 after a disastrous start to the 2015/16 campaign. Willian was one of the few star players during the last season and he also won the club's Player of the Season award.
Willian's impressive display saw him being rewarded with a new four-year deal worth £110,000-a-week ($142,120 per week) last summer. The Premier League leaders are determined to hold on to their star player, who has been a backup for Eden Hazard and Pedro for the majority of this season.
Mourinho tried to sign the 28-year-old in his first summer transfer window at United. However, the player revealed earlier this year that he snubbed advances from the Red Devils in order to continue at Chelsea.
"From Manchester United, because Mourinho is there too," Willian told Globo Esporte, when asked if there were any offers to leave last summer.
"From other clubs I do not know, but there are always some things on the internet. But I am glad to hear than there are clubs that admire my work."