Mendilibar paid the price for the club's dismal start to the season on Tuesday when he was fired after just eight matches in charge.
Levante, who have picked up just five points this term and sit second-bottom of the Spanish top flight, have now turned to the experienced Alcaraz.
The 48-year-old brought an end to his second spell in charge of Granada in May, having kept the club safe from relegation, and has also coached both Almeria and Recreativo Huelva on two occasions.
In addition, Alcaraz has taken charge of Dos Hermanas, Racing Santander, Xerez, Real Murcia, Cordoba and Greek side Aris Thessaloniki.
Alcaraz, who will be supported at Levante by assistant Jesus Canadas, is set to take charge of his new side for the first time on Friday when they travel to Celta Vigo.