The goalkeeper is flying to England on Wednesday night and is expected to have his medical on Saturday.
The Premier League club made an initial €70m bid for the Brazilian on Tuesday, which was rejected, but the clubs have now reached a deal.
Liverpool and Roma agreed the final details of the transfer on Wednesday with the fee, which include €5m in add-ons, to be paid over two years.
Chelsea were ready to move for Alisson if Thibaut Courtois departed but Liverpool have acted swiftly in an to secure their No.1 goalkeeper target.
Chelsea had, by Wednesday lunch time, not made an offer.