HITC Sport understands that the Cottagers have returned with a £4 million offer for the highly-rated defender.
The highly-rated 23-year-old is considered by many to be one of the finest centre-halves currently plying their trade outside of the Premier League.
Fulham have ambitions of gracing that top-flight stage again in the not too distant future and are busy trying to assemble a squad that will carry them back to that level.
Dunk figures prominently on their recruitment wish list, with HITC Sport bringing you news in June that a bid of around £2.5 million from the Cottagers had been knocked back by Brighton.
It was acknowledged at that time that Seagulls boss Chris Hughton is reluctant to part with a man who only penned a four-year contract in 2014, and one he considers to be a key figure for him heading forward.
Fulham, though, are refusing to give up the chase and are ready to test Brighton’s resolve once again.
Having upped their offer for Dunk, they will be hoping that the terms on the table will tempt a Championship rival into opening transfer negotiations.