Germany says it could take half a million refugees annually as Greek islands struggle to process a huge backlog of migrants desperate to travel to western Europe.
Reflecting deepening concern, the European Union's president warned on Tuesday (local time) the EU faced a years-long refugee crisis, while the UN urged countries worldwide to help.
German Chancellor Angela Merkel urged greater flexibility in EU migrant quotas as her deputy, Sigmar Gabriel, said Berlin "could surely deal with something in the order of half a million refugees for several years".
Germany, which has previously said it would take in 800,000 asylum-seekers this year, would continue accepting "a greatly disproportionate share" among EU members "because we are an economically strong country", Gabriel said.
But it was unacceptable for the EU to rely on just a few countries, such as Austria, Sweden and Germany, he added, stressing that European policy "needs to change".
On the Greek holiday island of Lesbos, where about 20,000 refugees and migrants have been waiting to be registered for travel to the mainland, a new processing centre set up overnight was helping to defuse a situation Athens warned was "on the verge of explosion".
About 30,000 people are stuck on islands in the Aegean Sea.
Greece said it had asked the EU for medical aid, bedding and over €9.5 million to support reception services on Lesbos, Samos, Kos and Chios.
Lesbos mayor Spyros Galinos said the pressure was easing after another 140 staff arrived from Athens to handle migrant and refugee registration.
A police source said on Tuesday night a staggering 14,000 people had been registered since the previous night. The huge queues seen earlier in the day had dwindled to just a few dozen.
Earlier, a handful of coastguards and riot police armed with batons struggled to control 2500 migrants in Lesbos's main port trying to board a government ferry bound for Athens.
They are desperate to get into northern EU countries that have thrown open their doors but between them lie several transit countries, some like Hungary who are are intent on trying to keep them out.
Several hundred migrants broke through police lines at Roszke, on Hungary's southern border with Serbia.
Further south, hundreds poured across Macedonia's border with Greece, another bottleneck.
The scenes underscored the task facing authorities from the Mediterranean to the Baltic Sea in handling the influx of people, many of them Syrians fleeing war and misery.
EU president Donald Tusk warns the situation may last "for many years to come".
The UN's special representative for migration and development, Peter Sutherland, called on Tuesday on countries around the world to join in the asylum effort, proposing an international conference.
European Commission chief Jean-Claude Juncker is expected Wednesday to unveil a plan to relocate 120,000 migrants from frontline EU states.
Germany would take more than 31,000 migrants, France 24,000 and Spain almost 15,000. British Prime Minister David Cameron says Britain can take 20,000 over five years.
The UN High Commissioner for Refugees says 381,412 people have arrived in Europe by sea so far this year, 121,000 of them landing in Italy and 258,365 in Greece.
A total of 2850 people have died or are listed as missing.