"Those without papers, the so-called illegal migrants, have the right to request asylum (or) the right to voluntary repatriation," junior interior minister for migration Yiannis Mouzalas told reporters.
"If they do not request either within 30 days, they will be returned to their countries of origin," he said.
Athens yesterday mounted a police operation to remove some 2,300 mainly African migrants from its border with Macedonia, where they had been blocked for days following the neighbouring country's refusal to allow economic migrants to continue their journey to Europe.
Other former Olympic facilities around the capital have also been used in rotation to house migrants since the crisis began earlier in the year.
A day after their arrival, officials said hundreds of migrants were already leaving the taekwondo hall for central Athens, hoping to come into contact with smuggling networks and continue to western and northern Europe.
"I have family in Denmark," one migrant told state television ERT outside the hall.
Asked how he planned to get there, he replied: "I don't know".
Hit by economic crisis for the past five years, Greece has struggled to manage the unprecedented influx of exiles -- mainly refugees from war-torn Syria.
The UN says migrant and refugee arrivals in Europe for 2015 could top one million by year's end.
Greece has called for additional EU assistance in funds and manpower to better screen the migrants and root out possible extremists.
At least two of the participants in last month's jihadist attacks in Paris came through Greece posing as Syrian refugees.
Athens recently attempted to return some 50 Pakistanis but Islamabad accepted only 20 of them, Prime Minister Alexis Tsipras said this week.
"We have asked Turkey to honour readmission agreements and have asked the same to happen with... Pakistan and Morocco," Tsipras said.
"Greece cannot become a warehouse of souls for people who don't want to stay here," he said.
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