The versatile defender has impressed for the 'Royals' since joining the club from Leicester in 2016, becoming a key part of the side that reached last season's play-off final.
However, a disappointing 2017/18 campaign that saw the club finish 20th in the table has led to a number of stars wanting out.
Moore, who can play at both centre-back and right-back, was ever-present for the club in the Championship this season, playing every minute of their 46 league games and attracting the interest of clubs further up the food chain.
Brighton are thought to be just one of a number of sides to hold and interest in the 25-year-old, with the aformentioned source reporting that they have followed up their interest with an official bid.