The EU and Turkey have so far failed to agree on a way to ease the migrant crisis during a summit in Brussels.
They are discussing an offer by Turkey to do more to stop illegal migration in return for more EU aid and accelerated EU membership talks.
An EU official said "good progress" had been made but EU leaders needed more time to study the Turkish proposals.
Europe is facing its biggest refugee crisis since World War Two, with most migrants coming via Turkey.
Turkey is currently sheltering more than 2.7 million refugees from the civil war in neighbouring Syria. The EU wants Turkey to take back migrants who do not qualify for asylum and do more to patrol its own waters.
In exchange, the Turkish government is reportedly asking for a doubling of EU aid aimed at helping Turkey deal with the crisis, to €6bn ($6.6bn; £4.64bn).
The Turks also want the EU to resettle some Syrian refugees currently on Turkish soil, as well as a faster path towards EU membership and the speeding up of plans to allow Turks visa-free travel in Europe.
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The latest proposal was put forward by Turkish PM Ahmet Davutoglu who said the package was designed to "strengthen Turkish-EU ties not only on the illegal migrants issue but also in all challenging issues".
But Hungarian Prime Minister Viktor Orban – who has taken a strongly anti-immigrant stance – vetoed the plan to resettle refugees in Europe, his spokesman said.
As the talks continued late on Monday, Martin Selmayr, chief-of-staff to European Commission President Jean-Claude Juncker tweeted: "Good progress in difficult European Council talks on solving the refugee crisis."
It is unclear whether a breakthrough will be achieved overnight.
Meanwhile Turkey's President Recep Tayyip Erdogan accused the EU of failing to deliver on the aid it has already promised.
"It's been four months," Mr Erdogan said in Ankara. "My prime minister is currently in Brussels. I hope he will return with the money."
Last year, more than a million people entered the EU illegally by boat, mainly going from Turkey to Greece.
Many migrants leave Greece in an attempt to reach northern Europe, but eight countries have introduced temporary border controls.
Some 13,000 migrants are currently stranded in northern Greece, after Macedonia closed its border 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.
Ahead of Monday's summit German Chancellor Angela Merkel said: "Today is about finding a lasting solution together with Turkey… trying to find a way to stop illegal migration and improving the living conditions for the refugees."
UK PM David Cameron said there was "no prospect of Britain joining a common asylum process in Europe".
"We have an absolutely rock-solid opt-out from these things," he said.
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 and fewer than 700 migrants have moved.
The union may now overhaul its Dublin Regulation, which requires asylum seekers to lodge claims in their EU country of arrival, and instead adopt a centralised system for processing applications.
More than 2,000 migrants, mostly from Syria, Iraq and Afghanistan, continue to arrive daily in Greece from Turkey.
The Idomeni crossing on the Greece-Macedonia border has become the latest focus of the migrant crisis.
One camp resident, Syrian refugee Narjes al Shalaby, told AP news agency: "All we do here is sleep, wake up, sleep," she said. "We get hungry, we wait in the queue for two hours for a sandwich, we come back, we sleep some more."
Separately, Nato says it is expanding its naval mission against people-smuggling in the Aegean Sea to cover Turkish and Greek territorial waters.