Greece asked the European Union for aid on Monday to prevent it being overwhelmed by refugees, as a minister said arrivals on Lesbos had swollen to three times as many as the island could handle.
Its economy already stretched close to breaking point, Greece is struggling to cope with thousands of people, mainly from Syria, fleeing poverty and war.
Interim Migration Minister Yannis Mouzalas said 15,000 to 18,000 refugees were on Lesbos, an island he said could cope with 4,000-5,000. "The situation is wretched," he told state TV.
Tensions have flared on the islands of Lesbos and Kos, short boat journeys from Turkey where there are some 2 million Syrian refugees.
The International Rescue Committee said protests on the streets of Lesbos were putting the lives and safety of refugees stranded on the island at risk.
"We are truly in the midst of a humanitarian disaster," said Kirk Day, the aid agency's field director on the island.
He said many refugees had been stuck on the island for weeks, people were sleeping rough and hygiene was rapidly declining.
"None of these things can be addressed with this many people here ... The only way forward is to move these people off the island immediately," Day said.
Greece earlier asked the EU to activate its crisis-response body to provide staff, medical and pharmaceutical supplies, clothes and equipment, the Interior Ministry said.
It has also applied to the EU Commission for 9.6 million euros in emergency funding to bolster existing reception operations on the islands of Lesbos, Samos and Kos and in the Turkish border region of Evros, and to help set up a new centre on the island of Chios.
The health ministry said it had sent extra medical staff to Lesbos and Kos and extended the operating hours of health centres on the islands.
On Monday morning, the coastguard, aided by a passenger ferry, rescued 61 migrants whose inflatable boat ran into trouble off Lesbos.
A Greek ferry unloaded 2,500 migrants at the port of Piraeus, bringing the total number of people moved to the mainland since last Monday to more than 15,000.
Thousands more are waiting to be identified and ferried to Athens, from where most head northwest to other European countries, many via Greece's border with Macedonia.
Mouzalas said the government planned to move 12,000 refugees from Lesbos over the next five days by increasing the number of ships ferrying people to Athens from three to four or five.
It will also set up reception and registration centres and convert some ships into temporary homes for refugees, many of whom are sleeping rough on the islands' streets.
"We can't solve the problem but we can provide solutions to alleviate it," he said.
Earlier on Monday, Greece requested that the EU civil protection mechanism be activated "in order to substantially strengthen the efforts ... to manage a volatile situation".
The mechanism coordinates the bloc's humanitarian aid efforts, channeling aid and sending special teams with equipment to disaster areas. It has previously helped Greece fight forest fires.
On Friday, European Commission First Vice President Frans Timmermans and Migration Commissioner Dimitris Avramopoulos have already promised Athens 33 million euros ($36.8 million) to help it tackle the crisis.
The mounting refugee tragedy unfolding before our eyes is an alarm call, not just for Europe but for the United Nations. The international community needs to focus on how best to accommodate these traumatised and courageous people and identify and remove the causes of conflict exacerbation, so that hope can grow that their homelands can be made peaceful and habitable again.