United, resume their Europa League campaign in Wednesday, and go into the second leg of their round-of-16 tie against Austrian side LASK Linz with a 5-0 lead.
"We're four games away from winning a trophy," Solskjaer said at his pre-match news conference.
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"The team has developed all through the season, we're delighted with finishing third [in the league], but the next step is winning a trophy.
"The players want to play, they don't want a break. It's not easy to say 'no, you're resting.' I hope the players need to win like I do. For me we go into the tournament with players wanting to win, and we won't be giving them a break."
Speaking of his team's chances of beating LASK, he said: "We know it will be a difficult game, like it was last time — it wasn't a 5-0 difference.
"We know we have to perform and there is a chance for players to stake a claim for a place in the quarter-final, if we get through."

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United played the first leg of the tie back in March, before football was shut down due to the Covid 19 outbreak.
In the secind leg, they will be without defenders Axel Tuanzebe and Luke Shaw for the match at Old Trafford, but midfielder Jesse Lingard, who scored his first Premier League goal in 18 months in their final game against Leicester City, will start.
"Axel started getting niggles and had to have an operation in his foot. He'll be back early September. Same with Luke, he should be ready for when the [2020-21] league season starts," Solskjaer added.
"Jesse will be starting tomorrow, but the team won't be too different from LASK in March."
Solskjaer said he was delighted with the performance of defender Chris Smalling, who returned to the club after a season-long loan spell at Serie A side Roma.
"I think Chris has shown over his career at United how valuable he is," Solskjaer said.
"This season was about him being a regular. Now he's coming back and Chris has shown this season that he will be valuable."
Additional reporting by Reuters.
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