Signs in the stadium told fans the final had been delayed "due to the late arrival of fans at the stadium"
Uefa delayed kick-off by more than 30 minutes, citing "security reasons".
The match at the Stade de France in Paris did not start until 20:36 BST, with Real beating Liverpool 1-0.
Police outside the ground fired tear gas as a small number of supporters tried to climb over security barriers.
UK sports minister Nigel Huddleston tweeted: "We are very concerned about the upsetting scenes around the Stade de France last night and shall be working with the appropriate authorities to find out what happened and why."
Liverpool defender Andy Robertson said the organisation of the match was a "shambles".
Merseyside Police said its officers who were deployed in Paris and attended the match "reported the vast majority of fans behaved in an exemplary manner, arriving at turnstiles early and queuing as directed".
"Our officers returning today will conduct a formal debrief to ensure that we can fully support any subsequent investigation following last night's game," said assistant Chief Constable Chris Green.
Merseyside Police's matchday officers tweeted that it was "the worst European match I've ever worked or experienced".
They added: "I thought the behaviour of the fans at the turnstiles was exemplary in shocking circumstances. You were not late 100%."
Liverpool Mayor Joanne Anderson criticised the "appalling management and brutal treatment of fans" and said she would be writing to Foreign Secretary Liz Truss "requesting answers from Uefa and for [France's] President Emmanuel Macron to investigate".
Liverpool fans' group Spirit of Shankly said the situation was "totally shambolic and extremely dangerous."
Pictures had emerged in the build-up of huge crowds of fans queuing to get in.
Many claimed they had been at the stadium hours before kick-off but were stopped from getting into the ground.
"We are hugely disappointed at the stadium entry issues and breakdown of the security perimeter that Liverpool fans faced this evening at Stade de France," the Anfield club said in a statement.
"This is the greatest match in European football and supporters should not have to experience the scenes we have witnessed tonight.
"We have officially requested a formal investigation into the causes of these unacceptable issues."
On Sunday, Liverpool chief executive Billy Hogan said the club wanted supporters to come forward with their experiences to feed into any investigations.
"We want to provide those facts to the relevant authorities," he said.
"In the coming days, we will be setting up a mechanism for supporters to reach out to us directly to provide their experiences and accounts of what happened."
European football's governing body Uefa said it was "sympathetic to those affected by these events and will further review these matters urgently together with the French police and authorities, and with the French Football Federation".
"In the lead-up to the game, the turnstiles at the Liverpool end became blocked by thousands of fans who had purchased fake tickets which did not work in the turnstiles," it explained.
"This created a build-up of fans trying to get in. As a result, the kick-off was delayed by 35 minutes to allow as many fans as possible with genuine tickets to gain access.
"As numbers outside the stadium continued to build up after kick-off, the police dispersed them with tear gas and forced them away from the stadium."
French police had urged fans not to force entry with tweets 20 minutes before the game was scheduled to begin.
As kick-off approached, large areas of empty seats could still be seen inside the ground.
Outside, some fans were seen climbing perimeter fences or attempting to run past security. Riot police were also in attendance.
Liverpool players emerged to warm up again at 20:06, with messages on the big screens inside the stadium stating the delay was because of the late arrival of fans.
French police said fans with fake tickets had tried to force access to the stadium but "the rapid intervention of the police allowed the return to calm".
They added they were able to disperse fans "without difficulty".
They also said there were no major incidents at fan zones in the city, although 500 people were moved on from a bar near the Liverpool fanzone at Cours de Vincennes.
French sports minister Amelie Oudea-Castera tweeted: "The attempts at intrusion and fraud by thousands of English supporters complicated the work of the stadium staff and police but will not tarnish this victory."
While interior minister Gerald Darmanin also blamed British fans.
-- Courtesy of BBC Sport