The Spain international is said to have already agreed personal terms with Los Blancos over a summer move, two years after seeing a deadline-day deal fall through due to paperwork.
Stand-in stopper Sergio Romero was handed a rare league start in the goalless draw against Southampton on Wednesday night and will again be given the nod in next week's Europa League final against Ajax.
Mourinho describes De Gea as being the best goalkeeper in the world, however, and is hopeful of keeping hold of both his current keepers come the start of next season.
"David is top in the world and obviously we want to keep the top in the world," he told Sky Sports News. "I don't go with [Romero can replace De Gea] 'if he leaves' because I hope he doesn't.
"We need two top goalkeepers and a top young goalkeeper and this is what we have. David is David and Sergio is Sergio. Spain's national team goalkeeper; Argentina's national team goalkeeper; it's amazing to have it. I'm really happy with both of them."
Probed on when supporters can next expect to see the Spaniard in a United shirt, Mourinho added: "The first match against LA Galaxy in pre-season in Los Angeles. I hope to play Sergio in the final and hopefully we don't have problems with the keepers."
De Gea has featured 200 times in the Premier League for Man United, keeping 75 clean sheets during that time.