The Istanbul club are looking for a new coach after parting company with Roberto Mancini just nine months into his tenure.
Prandelli resigned in the aftermath of the Azzurri's World Cup loss to Uruguay - a defeat that consigned them, like England, to a disappointing group-stage exit.
"Galatasaray Sportif has announced that they have begun negotiating with the Italian football manager Cesare Prandelli," read a brief statement from Galatasary on Thursday.
"The public will be updated about further developments."
Former Everton and Manchester United manager David Moyes had been considered a candidate to take over at the Turk Telekom Arena.
A spokesman for Galatasaray confirmed last week that Moyes and the club's president had "exchanged ideas" but revealed that Germany boss Joachim Low and United States counterpart Jurgen Klinsmann were preferred options.
Latest target Prandelli, who had two years left on his Italy deal when he chose to step down, guided Roma and Fiorentina before replacing Marcello Lippi as head of the national side in 2010.
He led Italy to the final of Euro 2012, before securing a third-placed finish at last year's Confederation Cup.