But Gunners boss Arsene Wenger said "revenge" is not on his mind.
"We want to qualify and we want to knock them out. So is it difficult? Yes. Is it possible? Yes," said Wenger, whose side host Barca on Wednesday, 16 February before travelling to Spain on Tuesday, 8 March.
"Barcelona are certainly the favourites for the competition and a famous team but, on their side, they did not want us either because they know they will get a game.
"I think we are better than last year and we have a good opportunity to show that."
Barcelona are top of the Spanish league and recently defeated El Clasico rivals Real Madrid 5-0 - a performance Wenger described as
one of the best he had ever seen.
Tottenham have already travelled to the San Siro this season, where they lost 4-3 to Inter. The London club, playing in the competition for the first time, trailed 4-0 at the break before a sensational Gareth Bale hat-trick ensured a tight finish to the match.
"It is back to the San Siro," said Tottenham secretary Darren Eales. "Hopefully we can play like we did in the second half against Inter and not the first half.
"We will take confidence from the second half that we can compete with AC Milan. Gareth Bale has been in sensational form in the Champions League. We play attacking football and we do have a goal threat.
"Hopefully we will get an away goal and bring them back to White Hart Lane and sneak through."
Spurs won their group, which included FC Twente and Werder Bremen as well as Inter, and could have been drawn against Roma, Lyon, FC Copenhagen, Valencia or Marseille in addition to Milan.
"It will be a great game," said Tottenham boss Harry Redknapp of his team's round of 16 tie.
"I would have taken AC Milan before the draw. They are leading the league in Italy but it has the makings of a great game over two legs. I'm looking forward to it."
Tottenham travel to Milan on Tuesday, 15 February, with the return leg on Wednesday, 9 March.
Spurs are scheduled to play 10 domestic fixtures before their trip to Italy, and Redknapp said: "It's a way off. We can put it to one side and concentrate on the league for now.
"It's fantastic for us to be involved in the Champions League. Our first dream was to get into it. To make the group stage was great.
"Then to win the group in possibly the hardest group of all was fantastic."
Chelsea could have been drawn against the likes of Inter, AC Milan, Roma, Lyon or Valencia but they will face the first Danish side to reach the last 16 of the Champions League.
Football is football and anything can happen, no-one thought we could come this far
FC Copenhagen coach Sol Solbakken
They will travel to Copenhagen for the first leg on Tuesday, 22 February before the return leg on Wednesday, 16 March - and take on opponents who will be in the midst of a winter break that runs until March.
"We have not been beaten at home in the Champions League and we have played Barcelona and Manchester United," said FC Copenhagen coach Sol Solbakken.
"Our best chance is to play out of our skins to keep the second game interesting. Football is football and anything can happen, no-one thought we could come this far.
"It's very difficult for us but we are looking forward to playing a great team with a great history and it's good to have the first game at home."
Chelsea chief executive Ron Gourlay insisted that the London club would not be under-estimating their opponents.
"There's no easy games at this stage and we'll be very careful," said Gourlay.
"They've played 10 games so far in the Champions League with qualifying, and were second to Barcelona, so they're a very, very good side and well clear in their domestic league."
Marseille president Jean-Claude Dassier argued that his team would have a better chance against United than some of the other opposition in the draw.
"They're maybe the biggest club in Europe along with Real Madrid, historically speaking," said Dassier, whose team host United on Wednesday, 23 February before the return leg at Old Trafford on Tuesday, 15 March. "But I think the English will suit Marseille better than Barcelona's short passing game would.
"We'll need a strong defence against the famous Wayne Rooney and the others who play with him. It will be a good atmosphere at the Velodrome and I think we have our chances."
Manchester United defender Rio Ferdinand was a non-playing member of the England side that defeated Tunisia in Marseille at the 1998 World Cup.
British clubs' Champions League reaction
"It will be a hard place to go," said the United defender. "You are talking about passionate fans in a big stadium.
"This is the knockout stage and where you earn your pennies. It is not a league situation. There is no chance to make up ground. We have to be right on the day."
Lyon face Real Madrid and will be hoping for a repeat of last year's victory over their Spanish opponents at the same stage.
The French side won 2-1 on aggregate and chairman Jean-Michel Aulas said: "It will be a very different Real Madrid side this year and it will be a very difficult tie.
"They may want revenge for last year, when it was a great result for us."
Holders Inter defeated Bayern 2-0 in last season's final, which was played in Madrid.
"It is a very interesting draw," said the German club's director Karl Hopfner: "It is a very interesting draw to meet them again so soon after last season's final.
"We are looking forward to it but both of us have not started that well in the league or the Champions League."
Champions League draw
Roma v Shakhtar Donetsk AC Milan v Tottenham Hotspur Valencia v Schalke Inter Milan v Bayern Munich Lyon v Real Madrid Arsenal v Barcelona Marseille v Manchester United FC Copenhagen v Chelsea
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