Rodgers has increased his offer for the Brazil international after further negotiations with Inter over the past 48 hours and remains confident of striking a deal despite competition from Southampton.
Liverpool's latest bid is expected to be their last and they are not prepared to enter into a bidding war with Southampton, who have offered £10m.
But it is believed that Coutinho is not interested in a move to St Mary's and Liverpool hope to agree a deal before the weekend.
Liverpool's initial £6m bid was rejected late last week, but they have restructured their latest offer and agreed to pay an extra £2m up front. Inter have been beset by financial problems and could find the promise of £8m difficult to turn down.
Coutinho will require a work permit to seal a move to the Premier League. Earlier this week he had appeared to distance himself from a move to Anfield by insisting he wanted to stay and fight for his place, but Liverpool remain hopeful of completing their second signing of the transfer window.
Liverpool have already brought in Daniel Sturridge from Chelsea this month in a £12m deal and are interested in Blackpool's Thomas Ince, though not at the Championship club's £8m asking price. (© Daily Telegraph, London)