An Italian fighting alongside Kurdish forces in Syria against the Islamic State (IS) group has been killed, media reported Tuesday, citing the jihadists and his family.

"We have killed the Italian crusader", IS said in a statement released along with a photograph of Lorenzo Orsetti, 33, a former waiter and chef from Florence, according to the Corriere della Sera newspaper.

Orsetti's father Alessandro told the daily he and his wife were "very proud" of their son.

Orsetti had initially travelled to Syria in September 2017 as an observer but felt "increasingly involved", his father said.

His body "is apparently still in a combat zone. We hope to be able to recover it as quickly as possible," he said.

"Lorenzo once told us he'd love to be buried in Syria. We don't know what we'll do yet, but I think we'll respect his wishes".

Orsetti had left a "last testament" cited by La Stampa daily and other media.

"I have no regrets. I died doing what I believed was right, defending the weak," it read.

According to the Corriere della Sera, some 25 Italians have travelled to Syria since 2015 to fight alongside Kurdish forces.

Three men and two women are still there, it said.