Reports suggest that the transfer fee is in the region of £43.7 million.
The midfielder was heavily linked to Premier League rivals Manchester United over the last 12 months but the Merseyside outfit appears to have stolen a march on the Red Devils.
The Brazilian has been identified as a replacement for Emre Can, who is out of contract at Anfield this summer and has been heavily linked with a move to Juventus.
The 24-year-old would occupy a holding role in the middle of the park.
Fabinho spent two seasons on loan at Monaco from Portuguese club Rio Ave before making a permanent move to the principality in May 2015.
He has made more than 200 appearances for the club in all competitions and played a key role in their triumphant 2016/17 Ligue 1 campaign.
Welcome to #LFC, Fabinho. 🙌https://t.co/Xd7ylCfnqJ pic.twitter.com/nYbV243DY5
— Liverpool FC (@LFC) May 28, 2018
