The Seagulls' manager, Chris Hughton, has already signed two players ahead of next season, Mainz's Leon Balogun and Deportivo's Florin Andone, but is striving to continue adding to side.
Hughton is launching another bid for Moore having had their initial offer rejected. The defender joined Reading from Leicester back in 2016, and has featured in every league game for the full 90 minutes.
The Royals' campaign this season ensured that they avoided relegation from the Sky Bet Championship, but they only secured safety on the final day of the league.
Moore was also pivotal for the Foxes back in 2013-14 after they had won promotion back to the English top-tier and has experience with Leicester in the Premier League.