Liverpool battled back from a 3-0 halftime deficit to draw 3-3 with AC Milan after extra time, and then beat their Italian opponents on penalties in Istanbul on Wednesday.
But Liverpool face missing the chance to defend their title after failing to finish among the top four in the English Premier League.
UEFA spokesman William Gaillard said that Liverpool could only be admitted to the competition at the expense of another English team.
But the English Football Association has already said that Liverpool's city rivals Everton, who finished fourth, would keep their place in next season's competition.
"The rules are what they are," Gaillard told BBC radio, who said that a precedent had been set in 2000 when the Spanish football association allowed Real Madrid to defend their European title at the expense of qualifiers Real Zaragoza.
"They were used already once when Real Madrid won and were not among the qualifiers in the Spanish League. It is a tough decision to make but it is not for us to make, it is for the FA and they told us Everton would be the fourth English club."
Gaillard also told Sky Sports that he didn't expect Liverpool to play in next season's Champions League.
But UK sports minister Richard Caborn claimed on Thursday that it would be "a travesty" if Liverpool were not allowed to defend their title and said he had urged UEFA to re-consider their position.
Caborn told the UK's Press Association: "I have raised the matter with UEFA's chief executive Lars-Christer Olsson and said it would be a travesty if Liverpool were not allowed to play in next year's competition, particularly after such an amazing final.
"They should re-think this issue and I think they will have to. Nothing is set in stone and it seems only right that the winners should have automatic re-entry rather than having to qualify through the domestic system. It is only common sense.
"Brian Barwick (FA chief executive) and Geoff Thompson (FA chairman) were also there last night and they also expressed similar views."
Liverpool's manager Rafael Benitez said it would be "unbelievable" if his side were not allowed to defend their trophy, while chief executive Rick Parry claimed "the world would want to see us have the opportunity to defend it after that."
