The Reds have been locked in negotiations with the Serie A giants after having a £6.3m bid for the 20-year-old rebuffed on Friday, but are thought to have increased that to £8m.
However, they are now in danger of losing out on the Brazilian after new Saints boss Maurico Pochettino, who had him on loan at Espanyol last season, moved to bring the former Vasco da Gama youngster to St Mary’s
It remains to be seen whether Liverpool will respond with an improved counter offer although its thought the Anfield club are confident of striking an agreement given Coutinho’s supposed reluctance to move to the south Coast and are hoping that will see Inter lower their valuation.
With the Premier League transfer window closing next week Brendan Rodgers wants the deal completed before Monday and will possibly move for other targets if the Italians are unwilling to let the player leave the San Siro.
Rodgers is also expected to step up his search for a goalkeeper in the next seven days after Barcelona indicated their desire to sign Pepe Reina in the summer as a replacement for current number one Victor Valdes.