United were pegged back after going two goals ahead at Old Trafford and then conceded a last-minute equaliser as they were held 3-3 by Everton.
Dean Henderson is likely to be in goal for United's FA Cup tie against West Ham at Old Trafford on Tuesday night, and it will be interesting to see what happens if he plays well.
Not for the first time, Solskjaer was asked about De Gea's position and said: "Every player has to earn his place. We're a squad with competition and that applies to every single one in the team."
The United boss admitted it was disappointing that Everton had three shots on target and scored three goals.