Gonzalez was a stand-out performer for Costa Rica at the World Cup in Brazil as they made history by reaching the quarter-finals of the tournament.
The 26-year-old's superb performances at the World Cup alerted a number of clubs across Europe to Gonzalez, with the big central defender named in many pundits' team of the tournament.
Gonzalez is currently plying his trade in Major League Soccer after joining Columbus Crew from Norwegian side Valerenga in February.
Columbus Crew have been bracing themselves for interest in Gonzalez in the wake of his impressive showings in Brazil and Fulham's bid in the region of $3.5million (£2million) has been turned down by his club and MLS, as they are expecting higher bids for the player.
Fulham are in the market for new central defenders after seeing Brede Hangeland and Aaron Hughes leave this summer and they had made Gonzalez - capped 40 times by Los Ticos - one of their top targets.
A number of Premier League and other Championship clubs are also thought to be closely monitoring Gonzalez's situation, as they weigh up moves for the player before the close of the transfer window.