Migrant crisis: Britain set to accept more refugees
David Cameron is expected to announce plans later to increase the number of refugees being allowed into the UK.
The extra refugees are expected to come from UN camps bordering Syria, and not from among people already in Europe.
No specific figure has been agreed, but Mr Cameron has previously said the UK would continue to take in "thousands".
BBC political editor Laura Kuenssberg said Downing Street had found itself behind the curve amid public and media reaction to the humanitarian crisis.
The PM is likely to make an announcement in Madrid after talks with Spanish and Portuguese leaders that had been intended to cover Britain's proposals for EU reform.
Calls for the UK to take in more refugees have intensified after the publication of a picture of the body of a drowned three-year-old Syrian boy, Alan Kurdi, washed up a Turkish beach.
Mr Cameron said on Thursday that as a father he felt "deeply moved" by the image, but he has argued that taking on more people was not the simple answer.
But it now appears that his stance is shifting amid pressure from public and political figures, including:
Scottish First Minister Nicola Sturgeon has written to Mr Cameron calling for the UK to accept more refugees
Former Labour Home Secretary David Blunkett said the UK should take in 25,000 over the next six months
A petition calling on the UK to accept more refugees has got more than three times the 100,000 signatures needed for it to be eligible for a possible debate in Parliament
Meanwhile, a stand-off between police and migrants on a train in Hungary is continuing into a second day.
On Thursday, police let the migrants board the train in Budapest but then tried to force them off at a refugee camp to the west of the capital.
The prime minister isn't changing his argument.
He still thinks opening up Europe's borders and agreeing quotas will not solve the refugee crisis. In fact, he thinks it would make it worse by increasing pull factors and encouraging people traffickers.
But, as the crisis gets worse and the public and political pressure grows, the prime minister does now accept that Britain has a moral duty to do more.
While on a visit to Portugal and Spain to discuss his planned EU reforms, Mr Cameron is expected to say that he is looking at extending existing British and UN schemes that have so far brought a few hundred of the most vulnerable Syrian refugees to the UK.
No targets have been agreed and none is likely to be set today - although Mr Cameron has talked of taking thousands more - but he is unlikely to satisfy his many critics who want Britain to take in tens of thousands of refugees and who have been outraged by his reluctance to act.
Almost 5,000 Syrians have been granted asylum in the last four years and the UK has accepted 216 Syrian refugees under a scheme to relocate the most vulnerable begun in January 2014.
It is expected that this relocation scheme will be extended to see thousands more refugees brought directly to the UK from camps in Syria and neighbouring countries.
The BBC's Laura Kuenssberg said the prime minister would continue to make the argument that the problem also needs to be tackled at source.
Among those to add his voice to calls for the UK to do more is former Chief Rabbi Lord Jonathan Sacks, who told BBC Newsnight that the UK needed to make "a very clear and conspicuous humanitarian gesture".
"I think 10,000 is a figure that we could handle," he said.
"It's a figure to which Britain would respond - the churches, the religious groups, the charities, would all join in, local civic groups.
But UKIP leader Nigel Farage warned the European Union was sending a message that anybody who wants to can come to Europe which, he said, would compound the crisis by encouraging more people to make the journey.
"If we want to stop images like that picture of that three-year-old boy being taken out of the sea then we have to stop the boats from coming. That is absolutely vital," he told LBC radio.
Development charity Oxfam welcomed indications of a shift in UK government policy.
Chief executive Mark Goldring said: "Offering to resettle 10,000 Syrian refugees would bring the UK in line with other European countries who have already shown leadership in offering a haven to vulnerable refugees."
Former Conservative foreign secretary Sir Malcolm Rifkind told BBC Radio 4's Today programme that the UK should respond "in a positive way" to the crisis "and that must run into thousands of people being admitted to this country".
But he added that he had concerns, for Britain and any country taking people in, over "what signal that will send to those who are not in Europe at the moment".
He also said the obligation to help refugees should be considered by countries outside of Europe, pointing out that "very large numbers" went to areas such as the United States, Canada, South America and Australia following World War Two.