The Norwegian can move within touching distance of Champions League qualification with victory against West Ham on Wednesday night.
But he admits he is also expected to deliver silverware – and revealed his disappointment after FA Cup semi-final defeat to Chelsea on Sunday.
Solskjaer is adamant he did not prioritise a top four-finish, despite dropping key players in Paul Pogba, Anthony Martial and Mason Greenwood for the 3-1 Wembley defeat.
But the financial implications of missing out on the Champions League mean it is vital United return to European football’s biggest stage after a season away.
Solskjaer has undertaken a major overhaul of the squad he inherited from Jose Mourinho and appeared to reference that when weighing up his first full season in charge.
There is also still the chance of a trophy in the Europa League.
Solskjaer said: “We can sit back and see after the League is finished and when the Europa League is done to see how we’ve done and then we can evaluate from where did we start to how did we end up.
“Of course at Man United you want to win trophies, that’s a given.
“But when you lose a semi-final - we’re that close - we’ve come one step further.
“Unfortunately, it wasn’t to be. I would have loved to have a cup final to look forward to in a couple of weeks.”
United could move up to third place if Chelsea fail to beat Liverpool – and a draw will take them to at least fourth after Leicester’s defeat to Tottenham on Sunday.
They face Brendan Rodgers’ side in the final game of the season, with Solskjaer calling on his players to prove they see out the job.
“We’ve given ourselves a great chance and it is, as you say, squeaky bum time,” he said. “I’m looking forward to it and I’m pretty sure the players are as well.
“We’ve given ourselves a very good chance with a great run. You can talk about it all the way back to October and that international break when we came back and should’ve won against Liverpool.
“From then on, we’ve been a very good team and more consistent.
“When you lose a game you’re hurt and disappointed,” he added. “You’ve got to focus on the next thing. Defeat is not the end of the world. It was end of the FA Cup dream and a final, but it’s business as usual and onto the next game.
“We’ve had so many setbacks as a club, lost games before and now we have to put on a performance.
“I think we’ve grown over the whole season and the added pieces of Odion (Ighalo), Bruno (Fernandes) and Paul (Pogba) and Marcus (Rashford) coming back is a strength for us. We are better as a group, more robust and resilient.
“The proof is in the pudding and we’ll have to see on Wednesday how they’re going to perform.”