The European border agency Frontex says it is deploying equipment and additional officers to Greece.
It comes as Turkey vowed to open its doors for migrants to travel to the European Union.
Greek authorities say 10,000 migrants are trying to cross over the land border from Turkey.
At least 500 people had arrived by sea on three Greek islands close to the Turkish coast on Sunday morning, police said.
Turkish President Recep Tayyip Erdogan has said his county cannot deal with the amount of people fleeing Syria's civil war.
His decision came after at least 33 Turkish soldiers were killed in air strikes in Idlib province there last week.
#Frontex is in close contact with Greek authorities regarding additional support we can provide. We are redeploying equipment and additional officers to Greece and closely monitoring the situation pic.twitter.com/vAPcvEN9Lt
— Frontex (@Frontex) March 1, 2020
Turkey has said funds promised by the EU to help it deal with Syrian refugees have been slow to arrive.
The Greek foreign affairs ministry has accused Turkish authorities of a "disinformation campaign".
In a tweet it said: "The reality: 10,000 people were prevented from entering Greek territory (all along Evros) from yesterday morning to this morning."
It also said 73 people were arrested for illegally entering the country.
Greek Prime Minister Kyriakos Mitsotakis is set to chair a national security council meeting later on Sunday.
Συνεχίζεται εκστρατεία παραπληροφόρησης από τουρκικές αρχές. Η πραγματικότητα: Από χθες πρωί ως σήμερα πρωί αποτράπηκε είσοδος σε ελλ.επικράτεια (όλο τον Έβρο) σε 10.000 άτομα. 73 άτομα χωρίς σχέση με Ιντλίμπ εισήλθαν παράνομα σε ελλ.έδαφος, συνελήφθησαν&σχηματίστηκαν δικογραφίες
— Υπουργείο Εξωτερικών (@GreeceMFA) March 1, 2020
The EU and Turkey struck a deal in 2015, which saw the establishment of an EU facility for refugees in Turkey.
The facility was seen as the answer to EU member states' call for additional funding to support refugees in the country.
It saw a fund of €6bn divided into two tranches.
The first is to fund projects running until mid-2021 at the latest - though most projects will finish earlier.
The second tranche is to fund projects running up until mid-2025.
Irregular migrants gather at forestland near the Pazarkule Border Gate to enter Greece at Turkey's border in Edirne | Image: DHA/ABACA/ABACA/PA ImagesThe priority areas include education, health, protection, basic needs, socio-economic support and municipal infrastructure.
The full operational budget of the facility was committed in November 2019.
It is to coordinate and streamline actions financed from the bloc's budget to enhance support provided to refugees and host communities in Turkey.
On Sunday, Frontex said it is in close contact with Greek authorities regarding additional support it can provide.
"We are redeploying equipment and additional officers to Greece and closely monitoring the situation", it added.
The agency supports EU member states and Schengen-associated countries in the management of EU's external borders and fighting cross-border crime.
