ATHENS (Reuters) - Greek authorities on Wednesday postponed hearings for eight Turkish soldiers who sought asylum after they fled Turkey following an abortive coup attempt, a case that has underscored lingering tensions between the two NATO allies.

The men - three majors, three captains and two sergeant majors - flew a military helicopter to the northern Greek border town of Alexandroupolis on July 16, a day after the coup attempt unfolded.

Claiming they fear for their lives, the men have sought political asylum in Greece. They deny being involved in the coup.

Turkey has sought their deportation, calling them "traitors" and "terrorist elements". That created a dilemma for Greece, which now has to decide whether to hold on to the men or risk irking Ankara. Athens has said it will deal with any asylum request swiftly.

Two of the men who appeared at the Central Asylum Service in Athens on Wednesday asked for a postponement of interviews to better prepare themselves, said one of the lawyers, Vasiliki Ilia Marinaki. The interviews were postponed, with the first due to start on Aug. 19.

"They are afraid to go to Turkey," Marinaki told Reuters Television. "They told me that they will definitely be tortured. They told me exactly 'we are going to beg for death, we are going to be dead anyway'."

Relations between Greece and Turkey have improved over the years, but they almost went to war over an uninhabited islet in 1996 and they remain at odds over territorial disputes and ethnically split Cyprus.

Last week, the men were handed a two-month suspended jail sentence on charges of entering Greece illegally. Nonetheless, they remain in 'administrative' custody.

Those who appeared at asylum offices on Wednesday were accompanied by police, and held t-shirts over their heads to conceal their faces.

Since the coup attempt Turkey has launched a purge of the armed forces and judiciary, rounding up thousands of people.

The eight men say they did not know a coup was under way and were obeying orders by their superiors to transport the wounded from the streets to ambulances, according to their lawyers.

They said they fled to Greece when their Black Hawk helicopter came under fire by police on the ground.

(This version of the story corrects final paragraph to read Greece, not Turkey)

(Reporting By Phoebe Fronista, writing by Michele Kambas, editing by Larry King)