"UEFA strongly condemns the abuse directed at Michael Oliver and his wife," European soccer's governing body said in a statement.
"We have been in contact with them to offer our support and we trust the relevant authorities to take action against those individuals who have behaved inappropriately both on and off social media."
Juventus, who had wiped out a three-goal deficit from the first leg, were furious when Oliver awarded a soft penalty for a foul by defender Medhi Benatia on Real forward Lucas Vazquez.
Cristiano Ronaldo converted the spot kick to make it 3-1 and send Real through to the semi-finals 4-3 on aggregate.
The BBC reported on Sunday that British police were investigating threatening text messages sent to Oliver's wife Lucy, whose Twitter account was also targeted with threatening and abusive messages.
The 33-year-old Oliver was bitterly criticised by some Juve players after the match, with goalkeeper Gianluigi Buffon saying the official was not prepared for such a high-profile game and should instead have been eating crisps in the stands.
Buffon, who was sent off for his furious protest, said that Oliver had a "garbage bin" instead of a heart.
On Saturday, Buffon said in an interview that he would repeat the comments all over again but in "more civil" language.
"I am a human being who puts passion, sentiment and anger into what I do," he said.
Former England forward Gary Lineker was among those who said that Buffon's comments were over-the-top.
"These vituperous attacks on Michael Oliver and his wife are repugnant," he said on Twitter.
"Big admirer of Buffon but his behaviour since has been inflammatory and out of order. Oliver was doing his job and hard to argue against his decisions. Understand the emotion but time to defuse it, Gianluigi."
(Writing by Brian Homewood in Berne; Editing by Ken Ferris)
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