De Gea made a huge mistake to gift Mason Mount Chelsea's second goal in their 3-1 FA Cup semi-final victory on Sunday.
It is by no means the first time the Spain international has made such a mistake and there have been calls for Solskjaer to leave the 29-year-old out when United face West Ham in the Premier League on Wednesday.
The United manager was in no mood to disclose whether De Gea will keep his place against the Hammers as United go in search of a victory that would move them into the top four.
Asked if he faces a crucial decision on who will start between the posts at Old Trafford, Solskjaer said during a virtual press conference: "That's your job to talk about and write about and my job is to prepare a team for West Ham.
"We're not gonna talk about individuals because no one here will feel we put more pressure on them."
He added: "This is not going to become a David de Gea press conference, we’re just going to stick together. David's mentally strong enough to know his job is to perform in training the next day and be ready for the games."
Solskjaer revealed that Eric Bailly is set to miss the encounter with David Moyes' side after being taken to hospital with a head injury suffered during the loss to Chelsea.
Captain Harry Maguire is expected to be fit despite clashing heads with Bailly on a painful afternoon for United in London, while Luke Shaw (ankle) is poised to return.
"Eric stayed down in London for all the checks, he's done all the checks," Solskjaer said.
"I didn't see him yesterday and he followed the [concussion] protocols that are needed. I don’t think he'll be ready for this game, I've not seen him this morning yet.
"I think everyone is available apart from that. We've just got to stick together, stay strong, two games to go and we’ve given ourselves a good chance (of finishing in the top four)."