Greece says it has stopped nearly 10,000 migrants crossing the land border from Turkey.
Regardless, Greek police say at least 500 people on seven boats have reached the Greek islands of Lesbos, Samos and Chios, where camps for migrants are already heavily overcrowded.
Turkey has vowed to open its doors to migrants to travel to the EU.
Turkish President Recep Tayyip Erdogan says it cannot cope with the crowds of people fleeing the Syrian civil war.
His decision came after at least 33 Turkish soldiers were killed in air strikes in Idlib province in northern Syria this week.
Greek Prime Minister Kyriakos Mitsotakis has called a meeting of his country’s National Security Council for later on Sunday.
The EU border security agency Frontex said it was on “high alert” at Europe’s borders with Turkey.
Turkey takes in 3.7 million Syrian refugees as well as migrants from other countries such as Afghanistan – but had previously prevented them from leaving for Europe under an aid-related agreement with the EU.
Erdogan, however, accused the EU of breaking the promises it had made in 2016 when Ankara agreed to secure the EU’s south-west border.
What is happening in Greece?
Almost 10,000 migrants were prevented from entering Greece on the last day, according to the Greek government.
Some migrants threw stones, metal poles and tear gas canisters as they were stopped at the border.
Seven boats brought more than 300 people to Lesbos, four arrived in Samos with 150 and two in Chios with a total of 70 to 80 people, a police officer told Reuters news agency.
Groups of migrants were also seen wading through a river on Greek soil at Kastanies.
Many blocked migrants were sent to Evros, an area along the Turkish border.
Greek Deputy Defence Minister Alkiviadis Stefanis accused Turkey of encouraging migrants to make this journey.
“Not only do they not stop them, but they help them,” he told Greek television station Skai.
On Saturday, clashes between migrants and the Greek police occurred after President Erdogan gave the all-clear for migrants to enter the EU.
The EU said it was helping Greece and Bulgaria – which also borders Turkey – to protect the bloc’s parameters.