A divided European Union on Wednesday unveiled a fresh plan to shake up its failed asylum policy and force countries to share the burden of its unprecedented migrant crisis.

Just days after Greece began expelling migrants to Turkey under a controversial swap deal, a top EU official admitted that the bloc's current system was "not working."

"We need to reform our European asylum system," Commission Vice President Frans Timmermans told reporters in Brussels, as the bloc battles its worst migration crisis since World War II.

The influx of more than one million migrants fleeing war and poverty has put the bloc's cherished border-free rules under severe strain and sparked sharp divisions among the 28 EU nations.

Under the bloc's existing rules -- the so-called Dublin system -- migrants seeking asylum must apply in the country where they first arrive and are returned there if they move to somewhere else.

But critics have slammed this as obsolete and unfair to Greece and Italy, where most of the 1.25 million Syrian, Iraqi, Afghan and other migrants entered the bloc last year.

The rules in any case fell apart in 2015 as Greece and Italy, overwhelmed by the crisis, simply waved migrants onwards to countries where they wanted asylum, like Germany.

"This is neither fair, nor sustainable," Timmermans said as he outlined two possible new approaches.

The first, dubbed "Dublin plus", would be to keep the existing system but add a "corrective fairness mechanism" to redistribute migrants from a member state grappling with a sudden influx of refugees.

A majority of countries support this option, a European diplomat told AFP.

A second, more radical, proposal would be to automatically distribute migrants across the EU based on member states' population, wealth and capacity to take in newcomers.

This option had the support of Germany and Sweden, which have taken in the lion's share of migrants, the diplomat said.

"Both options will provide much needed solidarity," Timmermans added.

But EU states have already struggled to implement an emergency scheme agreed last September to relocate 160,000 asylum seekers from Greece and Italy. Only 1,100 of these have been resettled so far.

The delays have been blamed on a range of factors -- from governments trying to filter out jihadists from among the refugees following the terror attacks in Brussels and Paris to a lack of housing and education.

But sceptics say political foot-dragging has also played a part.

On the Greek islands, tensions are running high after the first 202 migrants were sent back to Turkey on Monday.

Deportations have stalled since then as thousands of migrants filed last-minute requests for asylum.

A Turkish official said the next transfer "has been postponed to Friday" at Greece's request.

The EU and Turkey struck a deal last month under which all "irregular migrants" arriving in Greece after March 20 face being sent back.

Under a "one-for-one" deal with Turkey, for every Syrian returned, another Syrian refugee will be resettled from Turkey in an EU country, with numbers capped at 72,000.

Human rights watchdogs say the scheme is badly flawed, and on Tuesday the UN refugee agency (UNHCR) said it was concerned for 13 people who may have been unable to register for asylum before they were deported.

The EU-Turkey deal, as well as a series of border restrictions along the Balkans migrant route, appear to have sharply reduced the number of new arrivals in recent weeks, and Germany said it could lift its controls on the frontier with Austria if the trend continues.

"We would not extend the border controls beyond May 12 if the numbers remain this low," Interior Minister Thomas de Maiziere said Tuesday.

Pope Francis will turn the spotlight on Europe's handling of the crisis with a visit next week to Lesbos -- part of the Greek island chain where hundreds of thousands of people arrived last year.

The pope, accompanied by the head of the Eastern Orthodox Church, the Ecumenical Patriarch Bartholomew, and Greek Prime Minister Alexis Tsipras, will visit Lesbos on April 14 or 15, a government source in Athens said.