The offer is a huge £53m up front, with £9m of add ons.
Roma are believed to want £66m, but the Reds are confident a deal can be done.
The Brazilian has long been Jurgen Klopp's number one goalkeeping target, but the Reds were put off earlier this summer by an asking price in excess of £70m - together with indications that Allison favoured a move to Real Madrid.
The move was also complicated by the astonishing success of Liverpool's last capture from Roma, forward Mo Salah.
The Italian club believed they had sold Salah too cheaply and were not prepared to lower their asking price, but the Reds' pressing need for a goalkeeping reinforcement has led to them testing Roma's resolve.