Earlier this month, Sunderland lost their place in the Championship, meaning that the Black Cats have now suffered back-to-back relegations and will spend the next campaign in League One.
At an awards night on Wednesday, Coleman said: "The fans were remarkable, and we didn't exactly deserve anything from them. I said how I wanted to come and manage Sunderland. I never said I want to manage in the Championship, I want to manage Sunderland football club, and that is where I am.
"It has not gone the way we wanted it to go but I am still here and I would like the opportunity to put things right.
"That won't be my decision - whose it is I don't know. I think it is a super football club. It is a task with a lot of negativity but it is still a great football club."
Coleman has won just five of his 28 matches in charge of Sunderland.