Fulham have admitted that they may not be able to hold on to the midfielder, signed from Wigan in 2006, and the 29-year-old, who is sidelined with a groin injury, could be lured away by an improved wage offer, particularly as he has 18 months left on his Fulham contract
Megson has already been unsuccessful with bids for West Brom's James Morrison and Newcastle's Joey Barton and Bullard represents the type of player Megson wants to buy.
Arsenal were interested in the midfielder, however, and they are monitoring the situation while they wait to see if their attempts to sign Andrei Arshavin have been successful.
Meanwhile, former Arsenal striker Dennis Bergkamp has admitted that he could one day return to the club in a coaching capacity.
Bergkamp said: "I wouldn't rule out [working at Arsenal in some capacity]. I really love this club and the country and I love working here.
"I would advise everyone to work in England, as a player or a coach, it just breathes football. It's fantastic for the game."