The European Union has two months left to get its migration crisis under control or face the end of its cherished Schengen area, EU President Donald Tusk warned Tuesday.

"The statistics over the Christmas period are not encouraging, with over 2,000 [migrant] arrivals to the EU per day," Tusk told lawmakers at the European Parliament's plenary session in Strasbourg, France.

An EU summit on March 17-18 will be "the last moment to see if our strategy works; if it doesn't, we will face grave consequences such as the collapse of Schengen," he added.

The 26 countries in the Schengen area allow their citizens to travel freely across the bloc, without the need for passports or border checks. The free-travel zone has been in existence since 1985.

"The cost of non-Schengen is very high," European Commission President Jean-Claude Juncker told the parliament, pointing to the many cross-border workers and transport companies in Europe.

But some Schengen nations have already started re-introducing internal border controls in an emergency bid to stem refugee flows.

More than 1 million migrants risked their lives last year to reach Europe, many of them fleeing the war in Syria. The influx has created tensions between European countries and strained local resources.

While EU member states agreed to a raft of measures to tackle the crisis, they have been slow in implementing them.

A plan to redistribute 160,000 asylum seekers across the bloc has so far only led to 322 relocalizations, while a much-heralded deal with Turkey has yet to reduce the flow of refugees transiting through that country.

Europe's political system now hangs in the balance, Tusk warned.

"The EU is based on an assumption that ... member states are ready to act with goodwill and especially that they are ready to respect our common decisions," he said, quipping that neither he nor the commission has "tanks" to force governments into implementation.

International medical charity Doctors Without Borders (MSF) said earlier in the day that the EU and European governments collectively failed to tackle the refugee crisis last year.

MSF's report on European migration criticized governments for making the path for millions of refugees to Europe more difficult, thereby increasing risks along the way.

The year 2015 "will be remembered as the year in which Europe catastrophically failed in its responsibility to respond to the urgent needs of assistance and protection of hundreds of thousands of vulnerable people," the report said.

"The EU's deterrence and anti-immigration policies – developed over the last 15 years and further strengthened in 2015 – have increased the demand for migrant-smuggling networks and pushed people towards ever more dangerous routes which jeopardize their health and lives," it added.

The commission defended its role in tackling the migration crisis. Spokeswoman Natasha Bertaud said the EU's executive had been working since the start of the crisis to "drive forward a European and more unified" response.

Juncker laid the blame on EU member states, while also acknowledging that they are being battered from all sides.

"Europe is confronted with a convergence of multiple, complex ... crises, coming from the exterior or interior of the EU and happening all at the same time," he said. "We are clearly in a difficult situation."

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