The Barclays Premier League join Bolton, Blackburn Rovers and Wolves as sides understood to be watching the 19-year-old.
Fulton had previously been lined up as a potential signing by Wigan Athletic, in a deal which would have earned Falkirk around £250,000 in a development fee plus add-ons. The move was only cancelled after Owen Coyle was sacked as Wigan's manager.
Falkirk are aware that other clubs are interested, though. "It's not surprising a number of top clubs are seriously looking at him," said Alex Smith, the club's technical director.