Leaders of the European Union were aiming to sign an agreement with Turkey in Brussels on Sunday that offers Ankara cash and closer ties with the EU in return for Turkish help in stemming the flow of migrants to Europe.

"There has to be an agreement that allows us to implement an action plan between the EU and Turkey so that refugees can be received in Turkey, nearer their countries of origin, so that Europe can help Turkey, so that Turkey can live up to its commitments and borders can be controlled," said French President Francois Hollande.

Aware of a sense of desperation in Europe for a solution to a crisis following the arrival of close to a million people this year, Turkish President Tayyip Erdogan has been driving a hard bargain and diplomats said the 28 states had struggled through Saturday to agree a final offer.

The Europeans, and none more so than German Chancellor Angela Merkel, are under pressure to manage the biggest influx of people since World War II.

Measures the EU has taken in recent months have done little so far to control movements and while winter weather may lower the number for a few months, it is also worsening the plight of tens of thousands stuck by closing borders in the Balkans, piling further pressure on European leaders to find a solution.

Sunday's summit, called just days ago as Brussels tried to clinch a deal offered over a month ago, has been complicated by Turkey's downing of a Russian warplane on the Syrian border.

That has in turn complicated European efforts to re-engage with Moscow, despite a continued frost over Ukraine, in order to try and advance peace in Syria that could both end the flight of refugees and contain the Islamic State. The Islamist group's attack on Paris two weeks ago has also heightened public calls in the EU for more controls on people arriving from Syria.

By late Saturday, after a further meeting of EU envoys in Brussels, diplomats said a common EU offer that they hope will secure Davutoglu's signature was agreed. It should be approved by governments on Sunday before the summit.

There remains a degree of nervousness, however, that Turkey could hold out for more - a nervousness heightened by Erdogan's decision not to attend himself but to send his prime minister.

Cash for commitment

Europe wants Turkey to spend new EU money - some 3 billion euros ($3.2 billion) for the next year or two - on improving life for the 2.3 million Syrians now living in Turkey so that they are less likely to take to boats for nearby Greek islands.

The EU also wants Turkish authorities to make that journey more difficult and to keep out more of the Afghans and other Asians who cross Turkey on their way to Europe. It also wants to hold Turkey to commitments to take back people who successfully reach Greece but then fail in their claims for political asylum.

Turkey has pressed for more money. An original EU offer of 3 billion euros for refugee care over two years is now likely not to carry a clear timeframe as Ankara hopes to secure more.

Turks will be promised easier travel visas for Europe if they fulfill commitments on migrant flows in the coming year.

It will also win a pledge of "re-energized" talks on joining the EU, though with conditions to persuade EU states that do not want populous, Muslim Turkey ever to join, and also Cyprus, which wants Ankara first to help it reunite the island.

Diplomats said Turkey, whose decade-old EU accession talks have been stalled, would see talks on economic cooperation open shortly, but its demands for commitments from the EU to start talks on other issues such as human rights would remain on hold - satisfying Cypriot insistence it meet existing conditions.

Human rights concerns are unlikely to feature much in the talks on Sunday, diplomats said, despite calls from activists in Europe and Turkey for EU leaders not to ease pressure on Erdogan over issues such as media freedom because of the migrant crisis.