According to the Daily Mail, the Brazilian has grown frustrated at the lack of first team opportunities this season and is aware of Jose Mourinho's interest. The Portuguese signed Willian from Anzhi Makhachkala in the first season back at Stamford Bridge and the midfielder quickly established himself as a crucial figure for the west Londoners.
The Brazil international made 25 Premier League appearances in his first season in England, but featured in 36, 35 and 34 games in each of the next league campaigns, helping the Blues to two titles.
However, the 29-year-old has seen his opportunities drastically reduced so far this term, when he has started just eight of his 17 Premier League appearances and two of the six Champions League games he has featured in.
His lack of playing time has reportedly alerted Mourinho, who is eager to bolster his options out wide and had looked to sign a winger in the summer, only for Ivan Perisic to remain at Inter Milan after United failed to meet the Italian club's demands.
The Croatian has since signed a new deal with the Serie A leaders, meaning the United manager might have to look elsewhere.
Willian himself recently admitted that in the summer he held talks with United about a potential transfer as Mourinho was then on the hunt for a winger.
Despite the good relationship between Mourinho and Willian, however, it could prove hard for the Portuguese to lure the latter away from west London.
Chelsea manager Antonio Conte is understood to be keen to keep his squad intact in the January transfer window, given the Blues are still fighting on four fronts and face a hectic festive period.
Furthermore, earlier this week, the Brazilian himself appeared to rule out the possibility of moving elsewhere in the upcoming window.
"The fans can be calm [about me leaving]. I am a Chelsea player and I am very pleased with the way I have been playing in recent weeks," Willian said, as quoted by Evening Standard after scoring a goal in the Blues' 3-1 win over Huddersfield Town on Tuesday night (12 December).
"When I play I always try to do my best. I try to help the team to play well, to score goals and make assists. Against Huddersfield I did that with my team-mates."