The final match was held at the Stade de France in Paris on Saturday and ended with Real Madrid beating Liverpool with a score of 1:0.
"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," UEFA said in a Saturday statement.
According to UEFA, the build-up of fans led to a 35-minute delay of the game.
"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," UEFA said, adding that 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."
The Mirror newspaper said that French riot police attacked Liverpool football fans at a dedicated fan zone around Place de la Nation in the east of Paris on Saturday.
"Officers had been standing by with water cannon and other crowd control devices all day, and their numbers increased as the day went on. They seemed to be unhappy with the amount of drinking that went on, and so decided to move in. It seemed to be unprovoked," one onlooker told the newspaper.
There were reportedly women among the victims of the police violence, with law enforcement officers having attacked fans with tear gas, batons and shields, The Mirror said.
Brazilian winger Vinicius Junior scored the only goal in the 59th minute of the Champions League final.
Italy's Carlo Ancelotti, the head coach of Real Madrid, became the most successful manager in the tournament's history on Saturday, now having four Champions League wins on his record.