The 31-year-old's current deal is due to expire this summer, after which he will become a free agent.
Stockdale was instrumental in helping the Seagulls to win automatic promotion from the Championship this season, starting all but one of their games as they finished second in the final table.
However, the shot-stopper may not be with the club for their first top-flight campaign since 1983 having rejected the offer of a contract extension, according to Sky Sports News.
Stockdale joined Brighton from Fulham in 2014 and has gone on to make 139 appearances for the club across all competitions.
The Leeds-born keeper has previous experience of the top flight, having played 39 times in the Premier League during his six-year spell at Fulham.