After the 2015 migration crisis, a fence has been built along most of the border, and the European Union reached a deal with Turkey to prevent migrants from trying to reach Europe.
Those who reached the Turkish-Greek border were left stranded after both countries refused them entry.
"The government will do whatever it takes to protect its borders", government spokesman Stelios Petsas told reporters on Saturday.
The campaign came after Turkey pushed into Syria following the deaths of more than 30 of its soldiers.
On Saturday Turkish President Tayyip Erdogan said that some 18,000 migrants have already crossed borders from Turkey into Europe, and added that the number is bound to rise in the near future. He did not provide evidence of these numbers.
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"We are urging parties to allow civilians to move to safety, either in areas they control or across the front lines", said The International Committee of the Red Cross, and reminded the warring sides that hospitals, markets and schools are protected by law.
The statement added: "We are in close contact with Greek authorities regarding additional support we can provide in this rapidly evolving situation".
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Following these events, the EU's foreign policy chief Josep Borrell announced on Sunday an extraordinary meeting of the foreign affairs council would take place next week.
Greek's deputy defence minister said the country would not stand down.
The migrants are leaving Turkey through northwestern Edirne province bordering with Greece and Bulgaria.
The latest bout of fighting in Idlib has uprooted one million civilians since December, many of them women and children fleeing towards the Turkish border.
He underlined that the humanitarian crisis in the region will be deepened as the regime continues its attacks.
Around 3.7 million Syrian refugees are now residing in Turkey, which has repeatedly stated that it would not tolerate a further influx of Syrians to the country.
Greece said Sunday it has blocked almost 10,000 migrants at its border with Turkey, which opened its gates to Europe as tensions mount over its deepening conflict in Syria. Russian Federation has urged for calm from both sides, and on Saturday reported that Ankara had committed to reducing tensions on the ground.
The Greek government reiterated its promise to keep migrants out.