Liverpool want Uefa to "fully and transparently" fulfil recommendations made in an independent report on the events before last season's Champions League final in Paris.
Fans were penned in and teargassed outside Stade de France at the game against Real Madrid on 28 May 2022.
The report ruled Uefa bore "primary responsibility" for the chaotic scenes, adding it was "remarkable" no-one died.
Liverpool said action must be taken "to ensure there are no more near misses".
The club added the "fundamental safety failings" had "exacerbated the suffering" of the families, friends and survivors of the 1989 Hillsborough disaster.
"We implore Uefa to fully enact the recommendations as outlined by the panel - no matter how difficult - to ensure supporter safety is the number one priority at the heart of every Uefa football fixture," Liverpool said.
The report - commissioned by Uefa three days after the final which Liverpool lost 1-0 - made 21 recommendations in an attempt to ensure "everything possible is done" to prevent a similar incident happening at a major sporting event.
