The 23-year-old has been heavily linked with a move to Manchester United throughout this summer.
Jose Mourinho has already signed Victor Lindelof and Romelu Lukaku, but has made the defensive midfield a priority area for improvement.
The Portuguese boss wants to find a long-term replacement for Michael Carrick, who was handed a one-year contract extension and given the captain's armband following Wayne Rooney's departure.
He also wants to allow Paul Pogba to venture further forward and Fabinho is said to fit the bill with Chelsea reluctant to sell Nemanja Matic.
It appeared as if Manchester United would miss out on the Brazil international after an agreement was reached with Atletico Madrid.
United are said to have already had an opening bid rejected by Monaco for their prized asset, with it being £8.85million short of their valuation.
But Sport Review claims the Red Devils are ready to step up their pursuit of Fabinho by lodging an improved offer.
Fabinho, who can operate at both right-back and in central midfield, has supposedly already agreed on personal terms with United.
The former Real Madrid loanee admitted last month he would be "tempted" by a move to Old Trafford if a bid came in for him.
Fabinho told Brazilian TV station Esporte Interativo: “The offer would be tempting, first I would talk with my agents/representatives and with Monaco to make sure the right decision is made.
“But it's a great team and I would definitely think hard about the offer.”
Fabinho helped Monaco win the Ligue 1 title for the first time since 2000 last term.
He is contracted to the club until the summer of 2021.
Source: Daily Express