ANSA said Barenboim, 68, will assume the role left vacant by Riccardo Muti in 2005 on Dec. 1.
He is to continue in the position until the end of 2016.
La Scala Superintendent Stephane Lissner described Barenboim as "one of the greatest conductors of our time," ANSA said.
"The greatness of [Barenboim's] work, which I have had the opportunity to enjoy many times, will enrich the history of this temple to opera and the image of Milan and Italy in the world," said Milan Mayor Giuliano Pisapia, who is the honorary president of the La Scala foundation.
The seven-time Grammy Award-winner also is general music director of the Berlin State Opera and the Staatskapelle Berlin.
Celebrity Families of 2014 [PHOTOS]