Solskjaer was named boss in March.
United have won 15 games in all competitions – but stay in sixth place two points behind Arsenal, at the Premier League.
“We need players here who are likely to be able to simply take us forward,” he explained.
Asked about his squad, Solskjaer included:”It is likely to be survival of the fittest or survival of their best. That’s how ruthless we have to be.”
Solskjaer suggested that Wednesday’s 1-0 home defeat by Barcelona, at the first leg of the Champions League quarter-final, revealed far United have to really go to match the very best.
“We played with a Barcelona team which sets the norm in European football,” the 46-year-old told a news conference on Friday.
“We’re under no illusion it will likely be hard job and there will be some challenging decisions I made , Mick [Phelan, the assistant manager] and the club.
“You want to observe players step up and say now:’I need to participate in the.’ That’s what we expect from all the players.”
United’s group is set to change over the season. Contract talks with midfielder Ander Herrera are ongoing amid reports linking him with a move but Solskjaer confirmed Antonio Valencia will definitely be leaving and the club is going to be looking to get a new captain.
Considering who would take on the role, the manager said:”We have had Antonio, he’s going. Chris [Smalling] has been captain. You’ll find players here who can be captain on the pitch. We just have to choose who’s likely to be the voice in and around.”
United host West Ham at the Premier League on Saturday.