The Red Devils have not won the FA Cup since Louis Van Gaal led them to victory in the 2016 final against Crystal Palace.
Manchester United will welcome Wolverhampton Wanderers to Old Trafford later on Wednesday for a third-round replay after the first game at Molineux ended goalless.
United are off the pace for a first Premier League title since 2013, as they lie in fifth place, 27 points off leaders Liverpool.
United’s involvement in the Carabao Cup is also in doubt after they suffered a 3-1 first leg home defeat to Manchester City with the second leg to come at the Etihad later in the month.
They have a Europa League last-32 tie to negotiate with Club Brugge as the competition offers a back door route to the Champions League for the winners if they fail to qualify through their league position.
With the league title out of reach, Solskjaer is hoping to enjoy success in the cup competitions to bring back the winning feeling at Old Trafford.
"Supporters and the club alike aren't happy with not challenging for the top position in the league," the Norwegian said ahead of the Wolves tie.
"That's where we feel we should be and many of today's supporters have lived that period where we won the league consistently.
"At the moment we don't have that type of team, because in time we've been a bit behind the top ones.
"It might be that we start with winning a cup and then these players will get that taste. But that's not what we want to be.
"We want to be challenging for the league and the Champions League."
