A possible move to Everton has been touted, despite Gibson's insistence that he has no desire to leave United.
He has now admitted that he hopes to open contract talks with the Red Devils this summer, allowing him to prolong his stay at Old Trafford.
"My contract is coming to an end so, hopefully, at the end of the month I'll be signing a new one," said Gibson.
"My current deal ends at the end of next season but, normally, a player would enter into new contract talks at the end of this month.
"I've been told that a few clubs had expressed an interest in signing me, but I want to give it a good push and have a proper go at it next season before I decide to go anywhere else."