Performance-related add-ons could take the eventual fee to £18m.
Kelleher, a Republic of Ireland international, is someone Brentford tracked for a long time.
They were interested before they signed Mark Flekken in 2023, but Liverpool have only been open to selling since securing Giorgi Mamardashvili from Valencia last summer.
Brentford had been facing competition from other clubs for Kelleher, including Leeds and West Ham, but Thomas Frank’s side have a strong relationship with Liverpool, having secured Sepp van den Berg and Fabio Carvalho from Anfield last summer.
Kelleher, capped 22 times by his country, secured a second Premier League title with the Reds this season but only started 10 games as Alisson continued ahead of him in the pecking order.
