European Union leaders are holding a key summit with Turkey in Brussels on ways of dealing with Europe's worst refugee crisis since World War Two.
The EU is pressing Turkey, through which many migrants transit, to take some back in return for $3.3bn in aid.
Turkey is reportedly asking for that sum to be doubled. The talks are being extended into a working dinner.
Meanwhile Turkey's President Recep Tayyip Erdogan accused the EU of failing to deliver on the promised aid.
"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."
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 for this, the Turkish government is asking for the EU to increase its pledge to €6bn ($6.6bn; £4.64bn), European Parliament President Martin Schulz said.
Turkey is also seeking a faster path towards EU membership and the speeding up of plans to allow Turks visa-free travel in Europe.
Turkish PM Ahmet Davutoglu told reporters that he was proposing a "new package" designed to "strengthen Turkish-EU ties not only on the illegal migrants issue but also in all challenging issues as well as Turkey's EU accession process".
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- Disquiet over deal with Turkey
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.
After the talks started, organisers announced that the 28 EU leaders and Mr Davutoglu would discuss the latest proposals during a working dinner.
German Chancellor Angela Merkel said she anticipated a "difficult discussion", as a rift emerged among EU powers on the closure of the main route through the Balkans.
A draft summit communique reported by journalists on Sunday declared that "irregular flows of migrants along the Western Balkans are coming to an end; this route is closed".
But Mrs 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."
A German government source told the BBC there was no point saying the route was closed when 300-500 people still arrived in Germany every day this way.
Analysis: Chris Morris, BBC News, Brussels
This summit has hardly started and already it has been extended. There will be a second session with Prime Minister Davutoglu and all 28 EU leaders over dinner.
The EU is asking for a lot from Turkey, and Turkey is now asking for more in return. Hence the irony that the fate of the EU's migration policy, and its future cohesion, lies in the hands of a country it has been keeping at arms' length for years.
Without active Turkish co-operation, there is no chance of reducing the flow of refugees and migrants arriving in the Greek islands. And without that happening, the rest of the policy starts falling to pieces.
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.
EU leaders were also expected to raise the issue of the Zaman newspaper. On Friday, a Turkish court ordered the seizure of the opposition journal, increasing fears for media freedom. Two days later it was publishing pro-government articles.
'All we do is sleep'
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.
EU Commission spokesperson for migration Natasha Bertaud told the BBC all migrants rescued in Greek waters would be taken to a Greek island and screened.
All economic migrants would then be returned to Turkey to be screened again and "if they have no right to international protection" sent back to their country of origin.
All migrants rescued by Nato in Turkish waters would be taken back to Turkey.
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.