Turkey and EU to meet on worsening migrant crisis
"We will come together in Istanbul next Tuesday" with German Chancellor Angela Merkel and French President Emmanuel Macron, Erdogan told journalists on his plane back from Brussels, the Anadolu news agency said.
The French presidency did not confirm the summit when asked by AFP.
It follows talks between EU and Turkish officials on a migrant crisis, which has seen thousands of refugees head to the Greek border.
The two sides - which included President Recep Tayyip Erdogan - did not walk away with a firm agreement after "frank talks", which included an agreement from the EU to accept child refugees.
"There was a demand by the Turkish side to have a very close look at the 'statement' to see where we have and have not been successful ... this is a good start for further discussions," European Commission president Ursula von der Leyen said.
"We have different opinions on different things and that is why it is important to have a frank and open dialogue," EU Council president Charles Michel added.
Erdogan said it would be a four-party summit if UK Prime Minister Boris Johnson was able to join the talks.
The decision at the end of February to re-open Turkey's border for refugees seeking to reach Europe sparked a row with Brussels, as well as harsh exchanges with neighbouring Greece.
Turkey hosts some 4 million refugees - most of them Syrians - and is demanding greater assistance in dealing with the conflict in Syria and its humanitarian consequences.
So far, Europe has said it will consider taking 1,500 child refugees, but has focused more on reinforcing Greece's borders.
Erdogan sought support from both NATO and the European Union in Brussels on Monday, and despite apparent tensions in his meetings, he sounded upbeat on return.
"We can start a new process with the EU," he said. "We have taken many steps and will continue to do so."
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He hoped for progress on a new migrant deal with Brussels by the time of the next EU leaders' summit on 26 March.
Turkey agreed in 2016 to prevent migrant departures in exchange for 6 billion euros and talks on building closer ties.
But Ankara says it has not received all the money, and that other promises - including improved visa and trade rules - have not been met.
Meanwhile, Erdogan urged Athens to open its doors to refugees.
"This is my proposal to Greece: open the doors. Those people will not stay in Greece permanently and let them cross into other countries in Europe," he said.
Turkey re-opened the borders for refugees as it faced a potential new influx from Syria, where regime forces - backed by Russia and Iran - have been pushing to retake the last rebel province of Idlib.
Although Turkey and Russia agreed a ceasefire for Idlib last week, previous peace deals have proved temporary.
Erdogan said the ceasefire was currently holding.
"It's has been four days. I hope it will continue like this and turn into a permanent truce."
Foreign Minister Mevlut Cavusoglu confirmed on Tuesday that Turkey had requested Patriot missile defences from NATO. The issue has been complicated by Turkey's controversial decision to buy Russian S-400 missile defences.
Over 1,700 migrants have landed on Lesbos and four other Aegean islands from Turkey over the past week, adding to the 38,000 already crammed into abysmal and overstretched refugee centres.
The new surge has ramped up already high tensions on Lesbos, an island that has been on the migration frontline for years.
Frustration exploded into violence last weekend with mobs setting up roadblocks, attacking cars carrying NGO workers and beating journalists.
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Earlier on Saturday, the Greek migration minister announced plans for two new camps to house asylum-seekers who arrived after 1 March, when Turkey announced it would no longer prevent people from trying to cross into the European Union.
On the land border with Turkey, tens of thousands of asylum-seekers have been trying to break through for a week. There have been numerous exchanges of tear gas and stones with Greek riot police.
Far-right militants from other parts of Europe have travelled to Lesbos and the Greek border with Turkey, among them Swedish far-right leader Jimmie Akesson, who reportedly handed out flyers at Edirne with the message "Sweden is full".
On Friday, two Germans and two Austrians - identified as hardline nationalists by local media - told police they had been attacked and beaten on the central Lesbos market.
One of the four, who claimed they were journalists, was identified as Mario Mueller, a German member of the far-right Identitarian Movement.
On Saturday, anti-fascist activists organised a gathering in support of refugees on Lesbos.
"We need to react in some way because we've reached a point where fear is taking hold," said Maria Psomadaki, a retired teacher.