The Ivory Coast international, who had been given an extra week’s holiday, was due to show at Ciudad de Valencia before moving on to La Manga for a training camp.
But the 28-year-old’s absence adds fuel to reports he is angling for a move to the Premier League after hitting 15 Liga goals last season.
Kone was on loan from Sevilla last season, with a the deal allowing him to join Levante this summer.
But his contract also allowed Sevilla to invoke an automatic recall clause if he scored 18 or more goals, meaning he was rested for the last three games of the season at both his and Levante’s request.
Levante, who operated on a limited budget, are expected to sell him this summer, with Tottenham, Wigan, Reading and Aston Villa all linked with the former PSV forward.