Hernandez, 26, is out of contract in June, and Mundo Deportivo, citing Onda Cero, claim that Leicester have agreed to sign him on a free transfer.
The centre-back has made 30 La Liga appearances this season, and has been at Gijon since 2013.
It is claimed that Leicester have beaten off competition from Malaga and Real Betis to secure his signature.
The club appear to be planning behind the scenes to make new signings, and Hernandez is the latest name linked, with the deal claimed to be complete.
Leicester's signing of defender Yohan Benaoulane did not work out last summer, and he was sent on loan to Fiorentina in January, and Hernandez would potentially take his place in the squad next season.