The EU and Turkey say they have agreed the broad principles of a plan to ease the migration crisis at a summit in Brussels, but delayed a final decision.
European Council President Donald Tusk said all irregular migrants arriving in Greece from Turkey would be returned.
For each Syrian returned, Turkey wants the EU to accept a recognised Syrian refugee, and offer more funding and progress on EU integration.
Talks on the plan will continue ahead of an EU meeting on 17-18 March.
Europe is facing its biggest refugee crisis since World War Two. Most migrants come via Turkey, which is already sheltering more than 2.7 million refugees from the civil war in neighbouring Syria.
Read more about the migrant crisis
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- Disquiet over deal with Turkey
Turkey tabled new proposals ahead of the EU summit on Monday, and there was uncertainty on whether any agreement would be possible.
However, European Council President Donald Tusk said leaders had made a “breakthrough”, and he was hopeful of concluding a deal next week.
He said the progress sent “a very clear message that the days of irregular migration to Europe are over”.
In a statement, EU leaders said they broadly supported a deal that included:
- the return of all new irregular migrants crossing from Turkey to the Greek islands with the costs covered by the EU
- the resettlement of one Syrian from Turkey to the EU for every Syrian readmitted by Turkey from Greece
- speeding up of plans to allow Turks visa-free travel in Europe, with a view to lifting visa requirements by June 2016
- speeding up the payment of €3bn ($3.3bn; £2.2bn) promised in October, and a decision on additional funding to help Turkey deal with the crisis. Turkey reportedly asked for EU aid to be increased to €6bn ($6.6bn; £4.64bn)
- preparations for a decision on the opening of new chapters in talks on EU membership for Turkey
Speaking at a news conference after the summit, Turkish PM Ahmet Davutoglu said Turkey had made a “bold decision to accept all irregular illegal migrants… based on the assumption that for every one Syrian readmitted by Turkey from the Greek islands another Syrian will be resettled by Europe.”
But he said it was important to see the refugee deal as a package, to include progress on Turkish integration within the EU.
The BBC’s Chris Morris in Brussels says that, although this new initiative is bold, it could spark fierce argument and its implementation will not be easy. But, he says, the EU clearly needs Turkey’s co-operation if it is to begin coping with the migration crisis.
German Chancellor Angela Merkel said the proposals could be a major step forward if realised, stressing that “irregular migration” needed to be turned into “regular migration”.
French President Francois Hollande also hailed progress at the talks, indicating that aid to Turkey could be increased.
After the summit concluded, Luxembourg PM Xavier Bettel tweeted that Mr Tusk would “take forward the proposals and work out the details with the Turkish side before [the 17-18] March” migration summit.
UK PM David Cameron said EU leaders did have “the basis for a breakthrough”, which would mean that all migrants arriving in Greece could be returned to Turkey.
However before the summit, he stressed the UK would not take part in any resettlement scheme, saying: “We have an absolutely rock-solid opt-out from these things.”
Earlier a spokesman for Hungarian PM Viktor Orban – who has taken a strongly anti-immigrant stance – said he had vetoed the plan to resettle refugees in Europe.
Later on Tuesday, Mr Davutoglu will meet Greek Prime Minister Alexis Tsipras for talks in the Turkish city of Izmir.
Last year, more than a million people entered the EU illegally by boat, mainly going from Turkey to Greece.
Some 13,000 migrants are currently stranded in northern Greece, after Macedonia strictly limited the numbers allowed to pass through to all but a trickle.
The future of the Schengen agreement – which allows passport-free travel in a 26-nation zone – is on the agenda, as the leaders are anxious to save a system thought to bring billions of euros to Europe’s economy every year.
The EU said last October it would relocate 160,000 asylum seekers, mainly from Greece and Italy, but there was strong opposition among some members. Fewer than 700 migrants have moved.
Meanwhile more than 2,000 migrants, mostly from Syria, Iraq and Afghanistan, continue to arrive daily in Greece from Turkey.
Nato says it is expanding its naval mission against people-smuggling in the Aegean Sea to cover Turkish and Greek territorial waters.
EU leaders welcome the move on Monday, and called on Nato members to support the operation actively.