The 25-year-old is expected to undergo a medical on Thursday.
The Premier League club have landed Salah at the second attempt as three years ago they were outbid by Chelsea, with the winger moving to Stamford Bridge for £11m.
That move did not work out for him and, after loan spells at Fiorentina and Roma, he joined the latter for £13m 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 third behind Juventus and Napoli.