The defender looks set to leave Manchester City in the summer.
The Argentine is out of contract at the end of the season and looks set to leave the Etihad.
He has so far yet to agree a new deal, and by all accounts could be one of those to leave when Pep Guardiola takes over from Manuel Pellegrini.
Everton manager Roberto Martinez believes Demichelis would be a good influence on fellow Argentine Ramiro Funes Mori, while his defence could generally do with more experience.
That experience is also what attracts the Foxes, with them looking to boost their squad in the summer transfer window.
Claudio Ranieri wants more options to choose from next year, with his side set to play in the Champions League after a stellar campaign.
Both clubs face a battle for his signature though, with La Liga side Valencia also interested in his services.