The Brazil international was cast into doubt this summer with United, Barcelona and Real Madrid all keen on a transfer.
His late return to Chelsea training camp - caused by passport problems - sparked speculation that the player was unhappy at the club.
However, Willian told ESPN Brasil that, despite his strong relationship with former boss Mourinho, his plan is to remain in London.
Mourinho is the best manager I've ever worked with.
We have a good relationship, we are friends. Sometimes we talk, we text, we send messages to each other via WhatsApp.
He is a great manager, I really enjoyed working with him. I hope I can work with him again someday.
The 29-year-old was named as Chelsea's Player of the Year during Mourinho's ill-fated final campaign at Stamford Bridge and has regularly been linked with a move to Old Trafford since the Portuguese coach took charge of United in 2016.
Willian has previously suggested that United made an approach for him last summer, but is unsure if any formal offers have been made this summer.
I don't know if there was any official bid, but some things have happened.
Also, everything comes out on the internet, too. But Mourinho talks to my agent all the time, he says "bring him, bring him".
No, he has never called me, but he always sends me text messages.
But I am very happy at Chelsea, I want to stay in the club. I also like living in London, it is an amazing city, and my family is well adapted. So yeah, I intend to stay at Chelsea.