The Cottagers have scouted the teenage defender who is attracting attention from a host of top flight clubs.
Wolves have already made an enquiry for Murphy, who has yet to make his first-team debut for Pools.
Murphy, who has been on the bench a couple of occasions for Pools, has been watched by a number of Premier League clubs, including West Ham United and Wolves, who were reported to be scouting both him and Marcus Dinanga, on loan at the Super 6 Stadium from Burton Albion.
Fulham are believed to want to Murphy to go down and train with them with a view to completing a move for the promising youngster in January, although so far Pools are determined to resist offers for a player who has been on their books since he was eight years old.
New Pools manager Richard Money yesterday brought in versatile defender Harvey Rodgers from Accrington Stanley on loan.