Germany has called for a flexible system to allocate migrants as the EU prepared to announce a quota plan Wednesday, while Greek islands grapple with a huge backlog of people desperate to reach Western Europe.
European Commission chief Jean-Claude Juncker is preparing to unveil proposals for mandatory quotas that would relocate some 120,000 refugees across the bloc from border states that are swamped by the crisis.
German Chancellor Angela Merkel -- whose country expects 800,000 claims this year and has said it could take half a million refugees annually over several years -- said the plan "was an important first step," but cautioned against rigid ceilings.
"We need an open system to share out those with a right to asylum," Merkel said after talks with Swedish Prime Minister Stefan Lofven, whose country has also been more welcoming to migrants.
"There is another step that needs to be taken because neither Germany nor Sweden can determine the number of refugees" given the unfolding situation on the ground, she said.
Under the EU plan, Germany would take more than 31,000 migrants, France 24,000 and Spain almost 15,000. Australia, which has maintained a hard line on asylum-seekers, also pledged separately Wednesday to take in 12,000.
France has already agreed to take the 24,000 over two years and British Prime Minister David Cameron has said Britain can take 20,000 over five years, although opposition leaders in London have called for him to go further.
But mandatory quotas have faced stiff opposition from states such as Hungary that are on the front line of Europe's largest migrant crisis since World War II.
EU president Donald Tusk warned Monday that "the present wave of migration is not a one-time incident but the beginning of a real exodus" that would likely last "for many years".
- 'No bathrooms, no beds' -
Rights groups have criticised the EU response, which Amnesty International labelled "piecemeal and incoherent," and urged more support for border countries that are barely coping with the vast influx of people.
On the Greek holiday island of Lesbos, where around 20,000 refugees and migrants have been waiting in squalid conditions to travel to the mainland, a new processing centre was set up overnight to help with registrations.
Athens has warned the situation is "on the verge of explosion," while tensions are also rising across the Aegean Sea where around 10,000 more people are stuck on islands as they make their way to Western Europe.
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"It was horrible the last three days... There are no rooms, no hotels, no bathrooms, no beds, no anything," said Hussam Hamzat, a 27-year-old engineer from Damascus who finally got his departure papers Tuesday after an overnight wait.
Greece's migrant agency said it had asked the EU for emergency medical aid, bedding and over 9.5 million euros ($10.6 million) to support reception services on Lesbos, Samos, Kos and Chios.
More than 380,000 people have arrived in Europe by sea this year, new figures from the UN's refugee agency UNHCR showed Tuesday, including close to 260,000 who have landed in Greece and 121,000 in Italy.
Many are fleeing war and poverty in the Middle East, particularly war-torn Syria, as well as Africa and Asia in search of a better life in Europe, with most headed for top economy Germany.
- 'Every country in the spotlight' -
On Tuesday, the UN's special representative for migration and development, Peter Sutherland, called on countries around the world to join in the asylum effort, proposing an international conference "where every country is held up to the spotlight."
But in an illustration of the deep divisions in the EU's response to the crisis, Hungary vowed Tuesday to speed up construction of an anti-migrant fence on its 175-kilometre (110-mile) southern border aimed at stopping migrants.
Several hundred migrants broke through police lines at Hungary's main border crossing with Serbia, forcing police to use pepper spray to move a group off a main road, in the latest incident on its southern border.
Budapest has strengthened penalties for anyone found to cross is borders illegally after more than 165,000 migrants arrived this year, but the UNHCR's Europe director Vincent Cochotel warned the laws could "lead to chaos" when they are brought in on September 15.
The migrants' plight has touched hearts around the world, spurred especially by pictures last week of three-year-old Syrian Aylan Kurdi, whose lifeless body washed up on a Turkish beach.
Venezuela has said it would accept 20,000 refugees, Brazilian President Dilma Rousseff declared migrants would be welcomed there with "open arms" and Chile's leader Michelle Bachelet also vowed to take "a large number".
Canada's Quebec province has also said it will take 3,650 this year, and the US has said it is looking at how it could provide more help.