David Cameron is considering plans to admit thousands of unaccompanied migrant children into the UK within weeks, as pressure grows on ministers to provide a haven for large numbers of young people who have fled their war-torn homelands without their parents.

Amid growing expectation that an announcement is imminent, Downing Street said ministers were looking seriously at calls from charities, led by Save the Children, for the UK to admit at least 3,000 unaccompanied young people who have arrived in Europe from countries including Syria and Afghanistan, and who are judged to be at serious risk of falling prey to people traffickers. Government sources said such a humanitarian gesture would be in addition to the 20,000 refugees the UK has already agreed to accept, mainly from camps on the borders of Syria, by 2020.

Following a visit to refugee camps in Calais and Dunkirk on Saturday, Labour leader Jeremy Corbyn called on Cameron to offer children not just a refuge in the UK but proper homes and education, equivalent to the welcome received by those rescued from the Nazis and brought to the UK in 1939. “We must reach out the hand of humanity to the victims of war and brutal repression,” he said. “Along with other EU states, Britain needs to accept its share of refugees from the conflicts on Europe’s borders, including the horrific civil war in Syria.

“We have to do more. As a matter of urgency, David Cameron should act to give refuge to unaccompanied refugee children now in Europe – as we did with Jewish Kindertransport children escaping from Nazi tyranny in the 1930s. And the government must provide the resources needed for those areas accepting refugees – including in housing and education – rather than dumping them in some of Britain’s poorest communities.”

Signs that the prime minister may act came after a week in which concern has risen in European capitals, and among aid agencies and charities, about the high number of migrants still pouring into the EU just as cold weather bites along the routes many are taking through the Balkans and central and eastern Europe.

With one week of January to go, about 37,000 migrants and refugees have already arrived in the EU by land or sea, roughly 10 times the equivalent total for the month last year. The number of Mediterranean deaths stands at 158 this year. On Friday, at least 45 people, many of them children, died when their boats sank near two Greek islands, prompting German chancellor Angela Merkel to call again for a joint EU solution.

EU nations are, however, deeply divided over how to respond to the crisis, with several, including Germany, France, Sweden and Austria, setting up emergency border checks under special powers that expire in May. Last week Mark Rutte, prime minister of the Netherlands, which holds the EU presidency, said border-free travel under the Schengen agreement could break down within two months with devastating consequences for Europe and its single market unless solutions are found.

Aid agencies and charities are also reporting that the many messages coming out of EU capitals are creating confusion among migrants. Last week, after Austria placed a cap on the number of refugees it would take, aid workers said such announcements spread panic among refugees and made them more desperate to make their journeys before Europe’s doors closed.

Liberal Democrat leader Tim Farron urged the UK to admit more child refugees. He told the Observer: “Those who have made it to European shores now face cold winters, harsh conditions and are vulnerable to traffickers. We must open our hearts to those in need.”

Shadow Home Secretary Andy Burnham said: “ Such a gesture would endear Cameron to other EU leaders when he needs support in negotiations over the UK’s position in the EU. The prime minister would most probably get a better hearing from EU partners on his demands on free movement in advance of the referendum,” said Burnham.

Chancellor George Osborne, speaking at the World Economic Forum in Davos, said that the urgency of Europe’s refugee crisis was an incentive for Britain’s 27 EU partners to conclude a swift renegotiation agreement that would allow the government to propose a Yes vote in the up-coming referendum.