The Reds are understood to have already had one bid rejected by Inter, which was in the region of £6.75million.
Now the clubs are holding further discussions and the expectation is that a deal will be concluded shortly with Liverpool paying £8million.
The 20-year-old, who was linked with a move to Aston Villa earlier in the window, now looks close to becoming Liverpool's second signing this month.
Coutinho, who joined Inter from Vasco da Gama in 2010, spent last season on loan at Espanyol before returning to the San Siro in the summer.
He has featured five times for Inter so far this term.