A firefight erupted when the rebels were discovered crossing across the Syrian border into the town of Alakamis in the Sirnak province late on Wednesday, the local governor's office said in a statement.

The two sides opened fire after the four refused to surrender, it said, adding that the men were high-ranking members of the outlawed Kurdistan Workers' Party (PKK).

The PKK, listed as a terrorist organisation by Turkey and by much of the international community, took up arms in Kurdish-majority southeastern Turkey in 1984, sparking a conflict that has claimed some 45,000 lives.