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.
Rodgers, at his press conference to preview this weekend's FA Cup tie at Oldham, said: "We hope to do some business between now and the end of the month.
"The owners have been brilliant in supporting (me), we knew we were maybe a little bit short in the summer.
"We all agreed this was a month we would try to reinforce, if we could."
Coutinho came through the youth system at Vasco da Gama before joining Inter at the age of 16.
He spent the second half of last season on loan at Espanyol, where he worked with new Southampton manager Mauricio Pochettino.
Pochettino's recent appointment at Saints led to speculation linking him with the south coast club but it is understand Coutinho has expressed a preference for Anfield.
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