Press Association Sport understands negotiations have been finalised but, somewhat bizarrely, it is not yet known whether the transfer will break the club-record £35m the Reds paid Newcastle for striker Andy Carroll in January 2011.
That will depend on the exchange rate at the time the deal is formally concluded: 39m euros currently equates to about £34.3m.
Liverpool had an initial offer of £28million for the 25-year-old rejected earlier this month, but they have now come to a settlement having continued negotiations with the Serie A club.
The Premier League outfit have landed Salah at the second attempt as three years ago they were outbid by Chelsea, with the winger moving to Stamford Bridge for £11million from Basle.
That move did not work out for him and, after loan spells at Fiorentina and Roma, he joined the latter for £13million last summer.
The pacy attacker scored 15 Serie A goals last season - making it 29 league goals in two years - and provided 11 assists as Roma finished second, behind Juventus.
Source : PA
Source: PA