Prandelli submitted his resignation after failing to guide Italy out of the group stage at the 2014 FIFA World Cup in Brazil.
The announcement, which was made in the wake of the country's 1-0 Group D defeat to Uruguay, came despite Prandelli having signed a two-year contract extension prior to the tournament.
And Galatasaray confirmed in a statement on Thursday that they had entered into negotiations with the 56-year-old, who counts Fiorentina and Roma among his former clubs.
The Turkish giants have been on the lookout for a new coach since parting company with Roberto Mancini last month, and a move to Istanbul would represent Prandelli's first role outside his homeland.
Former Manchester United manager David Moyes was reported to be in talks with the club last week, but Prandelli has now moved into pole position to be appointed.
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