Dean Henderson has played the last seven games ahead of Thursday night's Europa League quarter-final clash with Granada in Spain, with the England keeper's selection for Sunday's win over Brighton particularly significant.
Henderson is now seen as United's new No 1, a situation that has cast doubt over De Gea's future after 10 years at Old Trafford.
Solskjaer admitted that neither of his keepers are happy at being left out, saying: "If it's Dean or David, I'm very comfortable with both of them to play and I'm very happy that none of them are happy to not play.
"David has been a top, top keeper and he still is. He's working to be ready when he plays.
"But every time I try to explain to players why they don't play in this particular game and what they need to do and what my plans are for them, all those discussions remain private.
"David has had 10 fantastic years at Man United and he's had challengers. Of course, they want to play.
"Sergio (Romero) probably played less games than he deserved, Dean has come in and given it his all because everyone wants that place in the team.
"David does as well and it's a healthy competition. The nature of football, goalkeeper departments are a very tight-knit group.
"They work together in prep, they have different gym sessions, individual sessions, they stay together more. Naturally they create a bond and the spirit in the goalkeeping department has been top.
"But it's a healthy competition. Both of them are important, all of them want to play. It's the same for Lee Grant and Nathan Bishop and Sergio Romero, they want to play and that's a competition I like."