The promising 21-year-old has been made available by Benfica due to an excess of midfielders in Jorge Jesus’ squad.
Record states that Monaco have made a proposal of €1.5m for a season-long loan with a €30m option.
However, Benfica are still not satisfied and are looking at a package worth over €40m for Florentino.
With three clubs interested, the Eagles will hope for a bigger offer before the transfer window closes.
The Portugal under-20 international is under contract until 2024 and shielded by a €120m release clause.
Florentino wants more game time than he had last season and a move is probable if the situation does not change.
He made 10 appearances in the league and two in the Champions League for Benfica last season.