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 currently moving positively.
talkSPORT understands that Leicester's initial £13million offer was deemed too low by Saints, who have told their Premier League rivals they require a deal worth £15m plus add-ons.
Denmark international Vestergaard is in the final year of his contract at St Mary's and will be allowed to leave if Southampton's valuation is met.
Saints have already allowed Danny Ings to leave for Aston Villa, meaning they would either have to act fast in finding a replacement before sanctioning a deal, or block a move until they can find a suitable replacement next summer.
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.