The news comes after recent reports that Leicester had made an initial bid for the Besiktas defender, but they have now raised their bid to around €14m, according to Turkish media outlet Sabah, after interest from Everton came to light.
Leicester are now the closer of the interested parties to meeting Besiktas' valuation of the defender, which is thought to be around €18m, but with his stellar performances at the World Cup so far, the price may not yet be set.
The centre back started in both of Croatia's first two games, including an outstanding performances in the shock 3-0 win over Argentina, keeping two clean sheets before being rested for the 2-1 win over Iceland, as Croatia had already qualified.
His impressive World Cup performances followed a season in which he starred for Dinamo Kyiv in Ukraine, before a January switch to Besiktas, where he made 18 appearances before being named in Croatia's 23 man World Cup squad by Zlatko Dalic.
From a Leicester perspective, Dalic would represent a fourth summer signing, and a third new defender, after James Maddison last week followed Jonny Evans and Ricardo Pereira in joining Claude Puel's new look squad.