In a statement on Friday, Peter Salama, UNICEF's regional director for the Middle East and North Africa, said that millions more people in Syria could become refugees and head to Europe if there is no end to the conflict that has bedeviled the country for the last four-and-a-half years.
Underscoring the scale of the challenge, a record 7,600 migrants entered Macedonia in just 12 hours overnight, according to a U.N. official. And Steinmeier said Germany expects some 40,000 migrants to arrive this weekend.
With the bloc continuing to squabble over a response, EU president Donald Tusk said he would call a leaders summit if a European justice and home affairs ministers' meeting in Brussels slated for Monday failed to yield a breakthrough.
"After contacts that I had with member states in the last few days, I feel more hopeful today that we are closer to finding a solution based on consensus and genuine solidarity," Tusk said. But "without such a decision, I will have to call an emergency meeting of the European Council," he said.
EU lawmakers have called for an international conference on migration bringing together the United States, the U.N. and Arab countries.
Facing criticism that his government has been too slow to help, U.S. President Barack Obama pledged to admit at least 10,000 Syrian refugees over a year starting Oct. 1.
White House Spokesman Josh Earnest said Obama had ordered staff to "scale up" the number after over 62,000 Americans signed a petition calling on Washington to take in more people.
The apparent failure of Steinmeier's mission to win eastern European states around to compulsory quotas came as record numbers of people — 70 percent of them fleeing Syria according to the UNHCR — entered both Macedonia and Hungary. In addition to the 7,600 entering Macedonia overnight from Greece, Hungarian police said 3,601 crossed the border on Thursday.
Some 432,761 refugees and migrants have crossed the Mediterranean so far in 2015 — more than doubling the total for all of last year, the Geneva-based International Organization for Migration said on Friday. The agency’s figures show an estimated 309,356 people had arrived by sea in Greece, with another 121,139 arriving in Italy, 2,166 in Spain and 100 in Malta.
Many of the refugees arriving in Greece head north across the border to Macedonia then travel on to Serbia and Hungary.
Hungary is, in turn, seen as a springboard to richer EU nations in the west and north, principally Germany and Sweden, via Austria, which on Thursday suspended rail services to Hungary.
The response of Hungary, which has seen some 175,000 migrants enter this year, has been to lay a razor-wire barrier along the Serb border and for almost 4,000 soldiers to begin erecting a fence 13 feet high with the help of prisoners from a nearby jail.
Meanwhile, new draconian laws that will come into effect on Tuesday will allow Hungary to jail migrants, and mooted legislation will see the army deployed and law enforcement given wide-ranging new powers.
Further concerns about Hungary’s treatment of refugees were raised Friday after video footage emerged showing migrants inside a holding camp being fed in the words of one volunteer "like animals in a pen," with women and children caught in a scrum.
"It was inhumane and it really speaks for these people that they didn't fight over the food despite being clearly very hungry," said Austrian volunteer Michaela Spritzendorfer, who filmed the scenes.
Hungarian Prime Minister Viktor Orban reiterated Friday that his country was merely applying European rules in seeking to register the new arrivals, pinning the blame on fellow EU member Greece for letting migrants leave and travel north.
"Just because Greece is not keeping to the common [Schengen] agreement does not authorize Hungary to give up on the Schengen rules as well," Orban said in Budapest.
Al Jazeera and Agence France-Presse