Ryan Lock is believed to have died during an offensive against so-called Islamic State forces to retake the northern city of Raqqa.
The 20-year-old, with no previous military experience, had reportedly joined Kurdish militia after telling his family he was going on holiday to Turkey in August.
In a statement Hampshire police said: “The Lock family are devastated by the recent news they have received but have no comment to make.
“The family politely ask that the media respect their privacy and leave them in peace to grieve.
“The family have provided a photograph of Ryan as they would like to remember him; a loving and caring family member.”
The People’s Protection Units, a Kurdish military force, are said to have told Mr Lock’s family that he had died along with four other fighters on December 21.