Manchester United will face the Spanish side Celta Vigo in the semi-finals of the Europa League over two legs and Ajax take on Lyon, after the draw on Friday.
Jose Mourinho's United will be strong favourites and have the advantage of playing the second leg at Old Trafford as they attempt to lift the Europa League, knowing victory in the final would give them a place in next season's Champions League.
United will travel to Spain for the first leg on May 4 before Celta visit Manchester for the return on May 11.
With United currently four points outside the Premier League top four, winning the Europa League for the first time may represent their best chance of qualifying for the Champions League.
Celta sit 10th in La Liga and progressed from the quarter-finals after following up a 3-2 home win against Genk by drawing Thursday's second leg 1-1 in Belgium.
Celta's top scorer Iago Aspas had an underwhelming spell at Liverpool, while fellow forward John Guidetti came through the ranks at Manchester City.
United have never faced Celta in UEFA competition and the Spanish side are through to a major European semi-final for the first time in their history.
As England's final European representatives this season, United were made to sweat by Anderlecht in the quarter-final second leg on Thursday.
Marcus Rashford's extra-time goal sealed a 2-1 win at Old Trafford that sent United through 3-2 on aggregate.
If United make it to the final in Stockholm's Friends Arena on May 24 it will be the club's first appearance in the decisive match of Europe's second-tier tournament.
It would also be United's first European final since they lost the 2011 Champions League decider against Barcelona at Wembley.
Having already bowed out of the FA and League Cups and with their Premier League campaign reduced to battling for a top four berth, United's only hope of winning a trophy in Mourinho's debut season is the Europa League.
Mourinho's men have been beset by injuries in recent weeks and they suffered more bad news on that front against Anderlecht on Thursday as their leading scorer Zlatan Ibrahimovic limped off.
United defender Marcos Rojo was also injured in the second leg to join Phil Jones, Chris Smalling and Juan Mata on the sidelines.
- Exciting challenge -
Despite their fitness woes, United are unbeaten in their last 22 league games and left-back Luke Shaw is confident they can finish the season with a European triumph in Sweden.
"They are three very good teams left in the competition, but we fancy ourselves and we will focus on ourselves," Shaw told MUTV before the draw was made.
"We'll focus on the two important games we have coming up in Burnley and Man City, take it from there and give everything.
"Obviously, we want to fight for the top four as well and now we have the semis and everyone wants to be playing."
United will be heavy favourites to advance, but Celta's sporting director Felipe Minambres says his club are relishing the chance to cause an upset.
"It's better to get things big," he said. "It's an exciting challenge against a great team.
"We are happy to be here and happier to have the chance to take on a team like Manchester United."
Minambres insists Celta will show United respect but he warned the Spaniards have enough quality to hold their own against their illustrious opponents.
"Our way of playing doesn't change at home or away. We have to enjoy an extraordinary match in Vigo," he said.
"United are a great team with a great name, but Celta has their strengths."