Gatti, 43, is the music director of Bologna's Teatro Comunale and London's Royal Philharmonic; Chailly, 51, left Amsterdam's Concertgebouw Orchestra last year and is to become general music director of Leipzig Opera and Gewandhaus Orchestra later this year.
Both have Milan connections. Chailly was assistant conductor at La Scala in the 1970s, and his father, Luciano Chailly, directed the company in the late 1960s; he is currently music director of the city's Giuseppe Verdi Symphony. Gatti is a native of the Milan and studied at the Verdi Conservatory there.
Antonio Pappano, the music director of the Royal Opera, has also been mentioned as a possible candidate, but columnist Norman Lebrecht reported before Muti's departure that Pappano is happy at Covent Garden and is "irritated" about the speculation. Others have proposed that the 71-year-old Claudio Abbado, who led La Scala from 1968 through 1986 and is much beloved, could return.