Despite the dangerous winter weather, nearly 37,000 migrants are reported to have arrived in Greece and Italy by sea and land this month.

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LONDON — At least 46 people — including 17 children — died when two wooden boats full of migrants trying to reach Greece sank early Friday, as a continued flow of asylum seekers put new pressure on European leaders to address a crisis that seems likely to intensify again come spring.

Authorities said the two boats had sunk near Greek islands in the eastern Aegean Sea as they carried migrants from Turkey toward Greece. The Greek coast guard rescued dozens and was continuing to search the area by sea and air, The Associated Press reported.

Greece is the most popular entry point into Europe for hundreds of thousands of migrants from Iraq, Syria and elsewhere. Despite the dangerous winter weather, nearly 37,000 migrants are reported to have arrived in Greece and Italy by sea and land this month, according to the International Organization for Migration, 10 times the number in the same period last year.

In the first three weeks of this year, at least 113 people died trying to make the sea crossing to Europe, the agency said, compared with 94 for all of January last year.

Despite the evidence that migrants from the Middle East and Africa are continuing to flee war and poverty in their home countries and will head out for Europe again in huge numbers this year, European leaders have taken no major new steps to curb the flow. They have also not agreed on a comprehensive border policy or prepared for another influx that could rival last year’s, when more than 1 million asylum seekers reached the Continent, many of them headed to Germany.

Prime Minister Manuel Valls of France issued a warning in an interview broadcast Friday about the future of a unified Europe, saying the idea was under threat unless the Continent could protect its borders.

Speaking to the BBC at the World Economic Forum in Davos, Switzerland, Valls said the Continent could not accommodate the enormous numbers of migrants and warned that they could destabilize European societies. “If Europe is not capable of protecting its own borders, it’s the very idea of Europe that will be questioned,” he said. “Europe has forgotten that borders are required.”

The Schengen zone, which permits largely unchecked movement across most of the Continent and was described by Valls as “one of the great European projects,” has been under severe strain as countries have introduced border controls aimed at stemming the flow of migrants.

Over the summer, Hungary erected a razor-wire fence on its border with Croatia. This month, Sweden, once one of the most welcoming of nations for refugees, introduced identity checks for travelers arriving from Denmark in an effort to curb their numbers. Austria said this week that it would limit the number of refugees it takes in this year to 37,500, down from 90,000 in 2015.

Chancellor Angela Merkel of Germany has come under increasing pressure at home and abroad for the country’s more welcoming approach to refugees. Valls, while acknowledging that she had “courage,” said, “The first message we need to send now, with the greatest firmness, is to say that we will not welcome all the refugees in Europe.”

Merkel met Friday with Turkey’s prime minister, Ahmet Davutoglu, the latest in a series of discussions between the countries about what more Turkey could do to keep the refugees pouring over its border with Syria from traveling on to Western Europe. Merkel again promised that the European Union (EU) would find about $3.2 billion to assist Turkey in offering work and schooling to more than 2 million Syrians who have fled there.

Davutoglu made no promises in return, making clear that Ankara believes it has leverage in negotiations on the issue with Europeans — like Merkel — who dragged out EU membership talks with Turkey in the past, in part because of concern over human-rights abuses in Turkey.

“We are not begging for the EU money,” Davutoglu told the German news agency DPA.