Modric is understood to be open to the move after initial discussions took place with the midfielder's representatives, according to Sky Italia.
The 32-year-old played in 43 games for Real last season, which culminated in their third straight Champions League title, and Modric has since captained his country to the World Cup final in Russia, winning the Golden Ball award for best player.
The Serie A giants managed by Luciano Spalletti are bolstering their squad ahead of the UEFA Champions League return this season and have shortlisted Modric as the ideal leader within their new-look midfield.
Reports in Germany also suggest that midfielder Arturo Vidal has also agreed a move to the San Siro from Bayern Munich.