The Merseyside club are thought to be prepared to pay £8.5million for the 20-year-old, who was also reported to have attracted interest from Southampton.
Manager Brendan Rodgers had said on Friday he was still looking to add to his squad with one or two new players before Thursday's transfer deadline.
Rodgers was unwilling to comment on speculation linking the club with Coutinho but it is now understand the club have succeeded with an approach after previous advances were rejected.
Coutinho is now expected to travel to Liverpool early next week for a medical and to complete other formalities. Liverpool will also need to apply for a work permit for the player.
If the deal is completed, Coutinho would become the club's second signing of this transfer window following the £12million capture of Daniel Sturridge from Chelsea.