(Adds detail on returns, EU official)

* Tusk says consensus emerging on migration crisis

* Commission sets out timetable for restoring open borders

* EU announces first refugee aid for Turkey, visa progress

* In return, EU wants end to migrant flow, more readmissions

* Merkel, Hollande say want to keep Schengen borders open

By Gabriela Baczynska and Paul Taylor

BRUSSELS, March 4 (Reuters) - European Union officials voiced guarded optimism on Friday that Turkey was starting to cooperate to stem the flow of migrants to Europe, as Brussels outlined a timetable for restoring open borders across the continent by the end of the year.

European Council President Donald Tusk, who will chair an emergency EU summit with Turkey on Monday, said after talks in Ankara he saw first signs that EU states were overcoming their differences to tackle the year-old crisis.

He also said Prime Minister Ahmet Davutoglu had told him Turkey was ready to take back all migrants apprehended in Turkish waters. The EU is demanding that Ankara crack down on people-smuggling and take back all illegal migrants from its shores who do not qualify for asylum in the 28-nation EU.

Even Syrians "apprehended" in Turkish waters, including by NATO patrols, would be put back ashore in Turkey.

"For the first time since the beginning of the migration crisis, I can see a European consensus emerging," Tusk said in a summit invitation letter to leaders.

"It is a consensus around a comprehensive strategy that, if loyally implemented, can help stem the flows and tackle the crisis."

The EU is trying to close its porous external borders and change the calculus of people fleeing war and poverty in the Middle East and beyond, offering them help if they stay put.

While Tusk was holding talks with Turkish President Tayyip Erdogan, the European Commission announced the first payouts from a 3 billion-euro ($3.3 billion) fund to help Ankara keep some 2.5 million Syrian refugees on Turkish soil.

It also said Turkey was making progress towards achieving eagerly sought visa liberalisation for its citizens in the EU.

EU envoy to Turkey Hansjorg Haber told reporters in Istanbul that 400 million euros had been disbursed on humanitarian aid and schooling for migrants.

After three months of mounting frustration in Europe since leaders signed a migration accord with Turkey, it was unclear what had moved Ankara to the point that an immediate reduction in flows seemed on the cards.

One EU official said Turkish leaders, busy with other priorities affecting their own national security, appeared to finally understand that Europeans might withdraw their various offers if numbers did not fall quickly.

Meeting in Paris, the leaders of Germany and France agreed that refugees fleeing war in Syria should stay in the region and said their common objective was to put Europe's frayed Schengen passport-free travel agreement back into operation.

"Our efforts are not done yet," Chancellor Angela Merkel told a joint news conference with President Francois Hollande. "I understand that Turkey also expects Europe to deliver."

Merkel pressed for Monday's summit with Davutoglu in an effort to demonstrate results before three regional elections in Germany on March 13 in which her conservatives face losses to the anti-migration Alternative for Germany party.

STAMPEDE

Tusk said Monday's summit would confirm the EU had closed the so-called Western Balkans route from Greece to northern Europe, which has been the main entry point for migrants.

"The number of illegal entries from Turkey to Greece remains far too high," he said after his talks with Davutoglu. Some 30,000 migrants are bottled up in Greece and more are arriving at a rate of 2,000 to 3,000 a day despite still wintry seas.

"We both believe that we can reduce the flow through the large-scale and rapid return from Greece of all migrants not in need of international protection," Tusk said.

On a visit to Athens, Turkish Foreign Minister Mevlut Cavusoglu said Ankara was seeing a significant decrease in the number of refugees arriving at its borders, due to its changing visa regime.

In Brussels, the Commission presented a step-by-step plan to implement agreed or already-proposed measures - including a new EU border and coastguard - to curb the influx after more than a million people arrived in an uncontrolled stampede in 2015.

"We cannot have free movement internally if we cannot manage our external borders effectively," Migration Commissioner Dimitris Avramopoulos told a news conference.

Eight countries in the 26-nation Schengen zone have put temporary, emergency border controls in place to control the flow of migrants, putting in jeopardy one of Europe's most prized achievements.

In a pre-summit report to EU leaders, the Commission estimated that a complete collapse of passport-free travel in the Schengen zone could cost the European economy up to 18 billion euros ($19.8 billion) a year. Much of the cost would fall on cross-border commuters, transport and tourism.

But investment bank JPMorgan Chase said the short-term impact of more probable selective border controls was likely to be "small in business cycle terms".

More than 1.2 million people submitted asylum requests in the EU last year, including 363,000 Syrians and 178,000 Afghans, the EU statistics agency Eurostat said.

Some 442,000 applications were submitted in Germany, the top destination for refugees and migrants, followed by 174,000 in Hungary, which has erected barbed-wire fences and used security forces to keep people out, and 156,000 in Sweden.

Sweden, long regarded as the most generous EU state towards refugees, said it would scrap payments of daily allowances to migrants whose asylum applications had been rejected, in its latest attempt to curtail the influx.

Fewer than one fifth of Germans believe the EU will agree on a common approach to the refugee crisis, according to a poll published by the daily Die Welt, and some 48 percent want Berlin to improve protection of Germany's national borders.

A clear majority - 56 percent - said Germany should cut its EU contributions if Monday's refugee summit fails.

While Brussels and Berlin are pushing for a European response to the crisis, more and more EU states are sceptical it could work and are resorting to unilateral steps.

"The Commission would never announce that Schengen is over," said one Brussels-based diplomat from an EU country.

"That would be a major political blow to them, the first real setback in the whole process of European integration. It would be like the pope announcing there is no God." (Additional reporting by Alastair Macdonald and Jan Strupczewski in Brussels, Paul Carrel in Berlin, Andrew Callus in Paris and Nick Tattersall and Humeyra Pamuk in Istanbul; Editing by Andrew Roche)