Brendan Rodgers wants to strengthen his backline with Wesley Fofana facing a long spell on the sidelines following a shocking challenge by Villarreal defender Fer Nino during a pre-season friendly last week.
Fofana suffered a fractured fibula and has since undergone surgery, but will not play until 2022 after the extent of his injury was ruled to be worse than first hoped.
As a result, the Foxes are prepared to delve into the transfer market once again this summer in the hope of finding a suitable replacement and Vestergaard could be the man.
The Foxes have remained in contact with Southampton about a possible transfer for the defender for the past year or so, with talks now at an advanced stage.
talkSPORT understands that Leicester's initial £13million offer was rejected by Saints, who told their Premier League rivals they required a deal worth £15m plus several add-ons.
Denmark international Vestergaard is in the final year of his contract at St Mary's, with Southampton prepared to sanction his exit as long as their asking price was met.
Vestergaard, who stands at a whopping 6ft 6ins tall, moved to St Mary's in the summer of 2018 and helped Denmark reach the semi-finals of Euro 2020 this summer.
West Ham were linked with a move for the centre-back, with The Telegraph claiming the Irons had identified the behemoth as an alternative to Kurt Zouma after the Chelsea defender rejected a move east.