Black Cats boss Gus Poyet wants to raid his former club for the 24-year-old midfielder and the ex-Chelsea trainee has tried to force the issue by asking for a move away from the Amex Stadium.
However, after originally refusing to comment, the Seagulls have confirmed that Bridcutt's request has been denied.
A statement on the Sky Bet Championship club's official website read: "Further to reports in today's media, Brighton & Hove Albion can confirm that a transfer request from Liam Bridcutt was received yesterday (Tuesday) afternoon and subsequently has been rejected by the club.
"At this stage the club does not wish to make any further comment on this matter."
Poyet has refused to go public with his targets for this January transfer window, but it is understood he is keen on completing a deal for both Bridcutt and his Seagulls team-mate Will Buckley.
The Uruguayan brought Bridcutt to Brighton in 2010 and he has gone on to make over 130 league appearances for the club.