Liverpool disappointed during the Champions League group stage
Liverpool will take on Romanian side Unirea Urziceni at the round of 32 stage in the Europa League.
Unirea, like Liverpool, dropped into the competition after finishing third in their Champions League group but defeated Glasgow Rangers and Sevilla.
Fulham face a tricky tie against Ukrainian side Shakhtar Donetsk.
Everton take on Sporting Lisbon, who play in the same city where David Moyes' team were defeated 5-0 by Benfica during the group stage.
Other notable draws include Ajax taking on Juventus and Hamburg playing PSV Eindhoven.
For us every competition and every game is important and the Europa League gives us an opportunity to win a trophy
Liverpool manager Rafael Benitez
Unirea are managed by former Sheffield Wednesday, Chelsea, Bradford and Southampton defender Dan Petrescu - and Liverpool boss Rafael Benitez has made it clear he will not be under-estimating the challenge.
"It will be a tough game because they were in the Champions League," Benitez told the club's website.
"I was following their group because I have some friends in Sevilla and so I know it will be difficult. But we are Liverpool and we have to have confidence.
"The main thing for me is that the game isn't until February so we have plenty of time to prepare and analyse the other team.
"We will be playing a side from another league and another level but winning their title means they are a good team. They went very close in the Champions League so we know it will be tough.
"For us every competition and every game is important and the Europa League gives us an opportunity to win a trophy. We will look to progress as far as we can."
Liverpool ambassador Kenny Dalgish was at the draw and suggested the club will be using their network of contacts to find out about their opponents.
"There's a phone call up the road to [manager] Walter Smith at Rangers, I'm sure he'll be happy to disclose any information he's got on them," said Dalglish.
"It will be a difficult game. They go on a break until March now but whether the rest helps them or hinders them remains to be seen."
Unirea general manager Mihai Stoica was delighted with the draw - but admitted that it will be a very difficult task for his team.
"For us it, is a great experience," said Stoica, whose club reached Romania's top division just three years ago.
"It is like David and Goliath. For sure, we will line up with 11 Davids and maybe it is not enough."
Fulham qualified from their group after a 3-2 win at Basle on Wednesday, but they face a stern test against the team that won the last Uefa Cup final in May.
Everton's opponents Sporting Lisbon are currently seventh in the league, 12 points behind leaders Braga.
The ties take place over two legs on a home and away basis, with all the first-leg encounters being played on 18 February except Everton, who will play two days earlier on Tuesday 16 February at 1745 GMT.
The Goodison Park club were forced to switch their fixture because both they and Liverpool have home ties in their first-leg matches.
Chief Executive Robert Elstone told the club website: "We battled Uefa extremely hard and made the obvious issues very clear indeed, trying every route to find an alternative. But the reality we were left with meant there was no other option."
Everton Club Secretary Dave Harrison explained: "In circumstances like this Uefa have a stadium and cities clash document which quite clearly indicates the procedures that need to take place and the order of priority for eligibility.
"Unfortunately, as a result of those priorities Liverpool have staked a claim for the Thursday night standard Europa League kick-off time.
"The same document makes it clear fixtures should be fulfilled on Tuesday with an 1800 CET kick-off (1700 GMT) if they cannot be played on Thursday. We have managed to negotiate an extra 45 minutes beyond that time."
Uefa broadcasting regulations mean that the game cannot be played later in the day as it cannot clash with Champions League fixtures.
Liverpool take priority over Everton as they finished higher than the Blues in last season's Premier League table. Had Everton won the FA Cup final against Chelsea, the situation would have been reversed and Liverpool would have been left with no option but to play on the Tuesday.
We have battled hard to get through the group stage and we were hoping for a good game, an interesting game.
All three English clubs play their return legs on 25 February.
Fulham manager Roy Hodgson believes his side have been handed one of the toughest Europa League draws by being paired with Shakhtar Donetsk.
The Ukrainian side beat Werder Bremen in the final of Europe's second competition last season before its name and format changed this summer and topped group J, which included Club Brugge, Toulouse and Partizan Belgrade to make the last 32 of the competition.
"It doesn't get much harder than playing the champions I suppose," said Hodgson. "We saw last year what a good team they are.
"Last year was a great success for Ukrainian football with Dinamo Kiev reaching the semi final as well so we couldn't have asked for a much tougher draw.
"On the other hand we have battled hard to get through the group stage and we were hoping for a good game, an interesting game."
He added: "I have just been looking at their squad and I think there are 13 foreign players, some of them Brazilians.
"They are a wealthy club. They buy players for a lot of money and obviously they are one of the top two teams in their league and they have also proved they are one of the top teams in European football."
The event took place at Uefa's headquarters in Nyon and included the draw for the round of 16 stage.
If Liverpool win their tie they will face the winner of the tie between Lille and Fenerbahce, Everton would play Atletico Madrid or Galatasaray, while Fulham would have a stern test against the winner of the tie between Ajax and Juventus.
Europa League round of 32 draw:
Rubin Kazan v Hapoel Tel Aviv Athletic Bilbao v Anderlecht FC Copenhagen v Marseille Panathinaikos v Roma Atletico Madrid v Galatasaray Ajax v Juventus Club Brugge v Valencia Fulham v Shakhtar Donetsk Liverpool v Unirea Urziceni Hamburg v PSV Eindhoven Villarreal v Wolfsburg Standard Liege v Salzburg FC Twente v Werder Bremen Lille v Fenerbahce Everton v Sporting Hertha Berlin v Benfica
Round of 16 draw:
Hamburg/PSV Eindhoven v Athletic Bilbao/Anderlecht Rubin Kazan/Hapoel Tel Aviv v Villarreal/Wolfsburg Atletico Madrid/Galatasaray v Everton/Sporting Hertha Berlin/Benfica v FC Copenhagen/Marseille Panathinaikos/Roma v Standard Liege/Salzburg Lille/Fenerbahce v Liverpool/Unirea Urziceni Ajax/Juventus v Fulham/Shakhtar Donetsk Club Brugge/Valencia v FC Twente v Werder Bremen
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