De Gea's current contract expires in 18 months, but United have an option to extend it for one more year.
But Mourinho is keen to avoid a replay of the transfer saga that almost saw the Spaniard leave for Real Madrid in 2015.
The 27-year-old is recognised as one of the best stoppers in the game and United are prepared to break the bank to keep him.
On De Gea's contract situation, his manager said: "It's obvious that we're not going to let the option disappear.
"A goalkeeper like he is, and obviously a club that wants to be better and better and better, we are not going to let that year option go away.
"But obviously we are going to try. [Executive vice-chairman] Mr [Ed] Woodward is not on holiday. He barely has holidays.
"Of course he's going to try to give him a contract that keeps him here for much longer than that option that of course we are going to execute.
"I just trust the board and the work they do. I don't negotiate players, I don't discuss numbers and contracts.
"I just say what is obvious. Any one of you would say the same -- David is to keep."
De Gea was rested for United's 3-0 FA Cup win over Derby last week.
But the former Atletico Madrid player is set to return to action tomorrow as the Red Devils face Stoke City (8.00pm).
Manchester United sit second in the Premier League, 15 points behind bitter rivals Manchester City.