United manager Jose Mourinho led his players in a solemn minute's silence on the club's Carrington training pitches to the west of Manchester city centre.
United were due to fly to Stockholm, scene of the final, later on Tuesday, but the club cancelled their pre-match press conference scheduled to take place at the city's Friends Arena.
"Due to the tragic events in Manchester last night and the effect that this has had on everyone here at our club and within our city, we have decided to cancel this evening's pre-match press conference in Stockholm," United said in a statement.
"We are sure that, in the circumstances, you will all understand our position on this matter.
"Our thoughts are with the victims at this terribly difficult time."
United's opponents in the final, Dutch club Ajax, tweeted: "From Amsterdam with love to Manchester. Our thoughts are with the victims and the loved ones of those affected."
Ajax head coach Peter Bosz was scheduled to hold a pre-match conference at the Friends Arena later on Tuesday.
The suicide bomb attack in Manchester, which claimed the lives of children and also injured 59 people, struck after a concert by American pop star Ariana Grande at the Manchester Arena.
It was the deadliest attack to hit Britain since the July 2005 attack on the London transport network, which left 52 people dead.
European governing body UEFA said it had "no specific intelligence" suggesting the Europa League final would be targeted by attacks.
It said security arrangements had already been beefed up following last month's truck attack in the Swedish capital, which claimed five lives and injured around 15 people.
UEFA urged fans to arrive at the stadium "as early as possible" on Wednesday as "detailed (security) checks will be made at the entrances".
UEFA president Aleksander Ceferin said: "I am deeply saddened by the horrible terrorist attack in Manchester last night.
"It shocks me that so many innocent people lost their lives and I would like to send my condolences to the families of those affected."
England's Football Association said "robust security measures" would be in place for Saturday's FA Cup final between Arsenal and Premier League champions Chelsea at London's Wembley Stadium.
It comes just over a month after the team bus of German side Borussia Dortmund was targeted by three roadside bomb blasts prior to a Champions League home game with Monaco.
That attack, which left defender Marc Bartra with a fractured wrist, is believed to have been carried out by a man who hoped to profit from a fall in Dortmund's share price.
United club captain Wayne Rooney expressed his shock on Twitter, writing: "Devastating news this morning. Thoughts and prayers are with all those affected."
Devastating news this morning. Thoughts and prayers are with all those affected.
Former United player Cristiano Ronaldo published a picture of a British Union Jack flag bearing the word "MANCHESTER" on his social media accounts.
"So sad to hear the news from Manchester," the Real Madrid superstar wrote on Instagram. "My thoughts are with the families and friends of the victims."
A help centre was opened at Manchester City's Etihad Stadium for those affected by the blast.
"The thoughts and prayers of our ownership, board and everyone at Manchester City are with the people of Manchester and all those affected following the horrifying events of last night's Manchester Arena attack," City wrote in a statement.
United are targeting victory in the Europa League final in order to complete their trophy collection and secure a place in next season's Champions League, having failed to qualify via the Premier League.
Ajax will be appearing in their first European final since losing on penalties to Juventus in the 1996 Champions League final.