The 19-year-old starred for the Cottagers during the early weeks of the season, contributing three goals and an assist in five Championship appearances.
However, at a time when he has less than a year remaining on his contract, speculation persists that Real Madrid are keen on signing the prospect in 2022.
According to The Sun, Fulham have offered Carvalho a "massive" long-term deal to continue his development in West London.
As it stands, that proposal has been turned down, and the second-tier side currently have no plans to put together another offer.
If Carvalho reaches January without penning fresh terms, Real Madrid or other foreign clubs will be in a position to hold formal discussions with the England Under-18 international.