The Barca captain, who scored his side's only goal in Sunday night's win at home to Granada in LaLiga to hand new coach Quique Setien a winning start, is left out of the squad for the cup clash.
"He has here on the beaches and in the restaurants in the summer," said Ibiza coach Pablo Alfaro ahead of the game. "It would be great to see him here in the winter as well."
But Messi will stay behind in Barcelona, along with Gerard Pique and Sergio Busquets.
The Catalan midfielder is suspended in the Copa del Rey from last season.
"I have always liked the Copa," Setien said on Tuesday. "We are going to try to win it, as well as the Champions League and LaLiga."
Youngsters Ansu Fati and Carles Perez, both regulars this season, are joined in the squad by Riqui Puig (following his impressive cameo on Sunday), Abel Ruiz and Chumi.
The new Copa del Rey format this year means ties will be played over one game and not two legs as in previous editions.