Moore has struggled for first-team football at Leicester during the last two seasons and Reading are hoping to lure the former England under-21 international away from the King Power Stadium.
Moore, who played a big role in Leicester`s 2013/14 promotion to the Premier League, found himself down the pecking order last season due to the form of the likes of Wes Morgan and Robert Huth.
The 23-year-old spent the first half of last season on loan at Bristol City and a number of clubs have made enquiries about signing Moore for next season if he does not figure in Claudio Ranieri's plans.
Rotherham United have shown an interest in taking Moore on loan, but Reading are hoping to win the race for his signature by making a move for the centre-back on a permanent basis.
Reading have opened talks with Leicester to see if they would consider selling Moore as new Royals boss Jaap Stam looks to bolster his defensive options ahead of the new season.