De Gea has developed into arguably the world's best goalkeeper at United having joined from Atletico Madrid in 2011, but his future has regularly been the subject of speculation.
Although United managed to tie him down amid interest from Real Madrid in 2015, that deal will expire in June if the club do not trigger its option for an extension.
The chances of United not doing so are minuscule, but that would only buy them another season, and rumours have suggested Paris Saint-Germain are interested in bringing De Gea to France.
He produced another remarkable save in Wednesday's (AEDT) 1-0 Champions League win over Young Boys when the game was level, and that prompted questions about the importance of securing his future.
"He's a world-class player," Mourinho said. "The best goalkeeper in the world and – if our ambitions are to be a big club, a winning club – you need the best goalkeeper in the world and also some other players who are the best in the world.
"In this case we have the best goalkeeper, and I know he wants to stay, I know his agent is happy to do what the player wants and I also know the board wants him to stay and they are working on that.
"Hopefully sooner or later they arrive into good conclusions."