Following De Gea’s early-season blunders, there have been calls for the young Englishman, 23, to replace the 30-year-old Spaniard.
But United manager Solskjaer believes it is not his job to keep them both happy.
He said: “It’s not about who has the most trophies.
“It’s about what you give to the team, here and now — we’re at Man United and we’re here to win.
“Everyone knows they have to perform to be in the team.
“We have at least two players in every position who’d feel, ‘I am a first choice’ and the keeper position is no different.
“It’s about your performances, so I’m now in a very fortunate position to have the goalkeeper department that I do.
“It was a conscious decision to bring Dean back to have real competition and he has made it harder and harder for me to leave him out.
“Whenever he plays, he does really well.”
The Norwegian added: “It’s not my job to keep any players happy, I’m not going round giving them games just to keep them happy.
“Of course we know we’re in a competitive environment. It’s about having the best possible chance to win and you need performances.