Scott Parker is looking to improve his Fulham squad after they won promotion back to the Premier League and has identified the 21-year-old as a target.
Record reports that Fulham want Florentino on an initial one-year loan with a mandatory purchase at the end of it.
This comes days after the same source reported that AC Milan are looking into a similar deal.
Milan’s offer would see them pay €3m, but for a two-season loan, then €30m to buy him permanently.
Fulham’s offer is reportedly around €1.5m as a loan fee, then €25m to purchase Florentino outright.
Florentino was an important part of Benfica’s side in the 2018-19 season, but an injury, then the arrival of Julian Weigl saw him drop down the pecking order.