The Spain goalkeeper, who almost joined Real Madrid in 2015 and continues to be linked with a move there, has a contract that expires in June 2019, although United have the option to extend it for a further year.
United manager Mourinho, though, says that the club are keen to negotiate a new contract soon, with British newspaper reports suggesting they would look to get him to commit to the club until 2023.
The 27-year-old, signed from Atletico Madrid for 18.9 million ($29.8 million, 22.7 million euros) in June 2011, was United's players' player of the year in 2014 and 2015, and Mourinho sees him as a crucial part of the club's bid to return to the very top of the English and European game.
"But obviously, we are going to try to give him a contract that keeps him here, for much more than the length of that option."
Asked if contract negotiations with De Gea were under way, Mourinho said: "I don't know. I just trust the board and the work they do. I don't negotiate with players, I don't discuss numbers in contracts, I just say what is obvious, and any of you could say the same, that he is to keep."
- 'I don't enjoy it' -
======================
United's manager has become involved in an increasingly bitter trading of insults with Chelsea head coach Antonio Conte over the past two weeks.
It culminated in Mourinho declaring that he had "contempt" for Conte on Friday, having earlier implied that he felt that the Italian was a "clown" because of the exuberant nature of his touchline celebrations.
Conte, for his part, accused the United manager of going senile, with Mourinho responding by referring to the four- month suspension that Chelsea's manager received in 2012 for failing to report match-fixing while at Siena. The Italian, who was cleared by a court of any wrongdoing in 2016, retorted by calling Mourinho a "little man".
"When I start (it), I take the responsibilities of that, and I've started many times. When I don't start (it), it's quite funny for me to see other people on the other side acting like victims when they are not the victims. But really, I don't enjoy it."
Mourinho pointed out that he apologised to Claudio Ranieri, his predecessor as Chelsea manager, after suggesting in 2008 that he had the mentality "of someone who doesn't need to win".
He said: "Sometimes, it's my fault. Sometimes, it's the other manager's fault.
"That's why our relationship went from bad to good, and from good to very good, because I was man enough to apologise.
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