The 19-year-old made a bright start to the new Championship campaign, scoring three goals and providing one assist in the opening five matches, before picking up a toe injury which has ruled him out for the last eight games across all competitions.
Fulham are at risk of losing Carvalho for free next summer as his contract expires in June 2022, and TEAMtalk claims that Southampton are one of a number of Premier League clubs who are monitoring the situation.
Fulham remain in talks with the player's representatives over extending his contract and they are confident that an agreement can be reached.
Southampton, meanwhile, believe they can persuade Carvalho to move to St Mary's and are prepared to table a cash offer in January to try and tempt Fulham to sell their highly-rated starlet.
Carvalho has played 11 times for Fulham's senior side since making his debut in September 2020, and has also featured on six occasions for the England Under-18s squad.