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Pecsi made the move to Liverpool from Hungarian outfit Puskas Akademia FC, who were in the initial rounds of the UEFA Conference League this season, only to be knocked out by eventual semi-finalists Fiorentina.
The fee was said to represent a record sale for his current employer.
Austrian-born Pecsi made 36 appearances in the season just gone as Puskas Akademia finished second (by three points) in the Hungarian top flight to Ferencvaros, a team managed by Robbie Keane and with former Liverpool midfielder Naby Keita on its books.
The 20-year-old is already a Hungary U21 international and has also represented them at U17 and U19 level.
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Armin Pecsi of Puskas Akademia FC
Armin Pecsi in action for Liverpool against Athletic Club
Every year, the Italian publication Tuttosport puts together the list of 100 players, and Pecsi is the second-highest ranked goalkeeper on there this time around.
The rankings will be updated every month until October and the winner will be announced in December.
Yamal and Pau Curbarsi of Barcelona are the top two players on the list at the moment, closely followed by Desire Doue of PSG
Pecsi comes in at number 47 on the list.
The new Liverpool goalkeeper is one of just three goalkeepers currently in the top 100