EU leaders in Brussels held “frank” talks with president Recep Tayyip Erdoğan on Monday over his decision to open Turkey’s border to migrants travelling to Europe, as deteriorating conditions in Greek camps led to the bloc agreeing to rehouse up to 1,500 child refugees.

The presidents of the European commission and council, Ursula von der Leyen and Charles Michel, battled during the difficult meeting in Brussels to save the current migration deal with Turkey in the face of Erdoğan’s claims that the EU had not kept its side of the bargain.

About 35,000 migrants have massed at Turkey’s borders with the EU in the last week since Turkey’s president broke a 2016 pact under which Brussels promised to pay €6bn (£5.3bn) in return for Turkey curbing migration flows.

Erdoğan has accused the EU in recent months of a lack of solidarity with his military operations in Syria and claims the true cost of housing refugees in his country has been close to €40bn.

As he greeted Michel, Erdoğan patted his chest with his hand and said “corona” in the direction of the assembled photographers, in an apparent explanation of the lack of a handshake between the two men. Michel said the two sides had “different opinions on different things, and that is why it is important to have a frank and open dialogue”.

Von der Leyen accused Erdoğan of politicising the border to extract concessions from Brussels and insisted that a “future-proof” solution should be found to avoid a repeat of the violent scenes.

“The events at the Greek-Turkish border clearly point to politically motivated pressure on the EU’s external border,” she said. “Finding a solution to this situation will require relieving the pressure that is put on the border.”

The two sides agreed that the EU’s high representative for foreign affairs, Josep Borrell, would work with his Turkish counterpart on “mapping” the differences on the migration deal.

Von der Leyen said they would “have a very close look with experts at the [2016] Turkey-EU statement”. Along with additional funds, Erdoğan is seeking progress on visa-free travel for Turks, and Von der Leyen said this would be among the issues addressed.

Erdoğan did not speak to reporters after the meeting, but in comments following talks earlier in the day with Jens Stoltenberg, the secretary general of Nato, he accused Greece of seeking to extract financial gains from the EU over the crisis.

“It is beyond reason and understanding that an ally and a neighbouring country points toward Turkey as the one responsible for the wave of irregular migration,” he said. “And I have expressed to the secretary general that we aren’t going to allow this country to use the current situation and use the EU in order to obtain unjust gains for itself.”

With the conditions at border crossings and in Greek refugee camps deteriorating by the day, the EU said earlier on Monday that it would take in between 1,000 and 1,500 unaccompanied children currently living in dire conditions.

Von der Leyen said five EU member states had come forward to offer help: Finland, France, Germany, Luxembourg and Portugal. “In the long term, we need a process to deal with unaccompanied minors, we need a systematic process,” she said.

She said the EU needed “to make sure that when unaccompanied minors end up in these camps that we have a mechanism to deal with them and a way to ensure they enjoy a safe future”.

Von der Leyen said the talks with Erdoğan in Brussels were only the beginning of a process to see how much further the EU could support Turkey financially in dealing with migrants arriving from war-torn states.

“The money has been paid or it has been contracted,” she said. “We ask ourselves how will the next step be, but this is the very beginning.”

Erdoğan announced last week that he would be “opening the doors” for refugees fleeing Idlib province, the final rebel stronghold in Syria.

Facebook Twitter Pinterest Recep Tayyip Erdoğan (left) meets Charles Michel and Ursula von der Leyen. Photograph: Anadolu Agency/Getty Images

Migrants and refugees have since clashed with Greek police, who have fired teargas and plastic bullets. Vigilante groups have started intimidating people at the border, with residents on the island of Lesbos reportedly blocking the disembarkation of a dinghy full of people travelling from Turkey, including a pregnant woman and children.

Erdoğan stirred the situation on Sunday by giving a speech in Istanbul calling on the Greek government to allow migrants and refugees to move on to the rest of Europe.

“I hope I will return from Belgium with different outcomes,” Erdoğan said. “Greece, I appeal to you … open the gates as well and be free of this burden. Let them go to other European countries.”

'Just run': on the Turkey-Greece border as refugees try to break through Read more

Johannes Hahn, the EU budget commissioner, suggested over the weekend that Erdoğan was using the crisis to distract from his weakened position at home. “It’s the standard reflex in response to all this. You seek an external opponent,” Hahn told the Austrian newspaper Der Standard. “Of course [the flow of refugees] is being steered.”

A former close Erdoğan ally became the latest to register a new political organisation to challenge the ruling Justice and Development party (AKP) on Monday, saying Turkey needed a fresh start. Ali Babacan, 52, a former deputy prime minister, quit the AKP last July, citing deep differences about its direction.

Another one-time Erdoğan ally, the former prime minister Ahmet Davutoğlu, established the Future party to rival the AKP in December.