Rodgers agreed to increase his offer after further negotiations with Inter over the last 48 hours and is confident of seeing off interest from Southampton, who are also interested.
Liverpool, who previously had a £6 million bid for the player rejected, are said to be prepared to walk away from the deal if their latest offer is not accepted, and will not enter a bidding war with Southampton, who have offered £10 million for the Brazilian.
Mauricio Pochettino, the new manager at Southampton, worked with Coutinho at Espanyol last season and the club have agreed to match Inter's valuation for the 20-year old, who is considered one of the hottest properties in South American football.
However, the player is not interested by a move to St Mary's and Liverpool hope to agree a deal before the weekend.
Inter have been beset by financial woes recently, and might find an £8 million offer too tempting to resist.
Coutinho will require a work permit in order to move to the Premier League and the latest haggling between Liverpool and Inter along with Southampton's lingering interest means it is unlikely that the player will arrive at Anfield in time for Sunday's FA Cup clash with Oldham Athletic.
Earlier in the week, we reported that Coutinho was set to snub a move to Liverpool and stay at the San Siro to fight for his place.
But Liverpool are confident of completing the deal and will make the Brazilian their second winter signing following the capture of Daniel Sturridge from Chelsea earlier this month.