United manager Jose Mourinho signed the Spain international whilst in charge of the Stamford Bridge club in 2014 and is reportedly keen on working with him at Old Trafford.
Fabregas, however, is keen to extend his stay in west London past his current 2018 expiry.
“Would I like to renew? Yes, because it is a club that has always treated me well," he told the Evening Standard.
“I believe the fans have a lot of respect for me and I have a lot of respect for them. I have given my all for the club for the last three-and-a-half years.
“This is a fantastic club for me and I have had a fantastic time here. Hopefully another contract will happen. Time will tell.”
However, Chelsea's policy to offer players over the age of 30 just one-year extensions means Fabregas' long-term future at the club will be under scrutiny after his current deal expires.
The former Arsenal man, though, insists he has plenty more to offer the club.
“I know that people panic after players reach 30 and stuff like that, but I don’t believe in this," he added. "I feel well, I feel young in my mind. I train every day as hard as possible.
“I think in the last 48 games I have played in 47, and I only missed one because I was suspended against Tottenham in August.
“I always try to give my best. I will play at any moment, even when I’m sick or not 100 per cent.”