David Cameron is expected to come forward with new plans within the next few days which could see thousands more Syrian refugees come to the UK.
Urgent work was under way in Whitehall on the scale and nature of the plan, amid intense pressure on the Prime Minister over Britain's failure to match other European countries such as Germany and Sweden, which have opened their doors to large numbers of people fleeing violence in the war-torn state.
Mr Cameron said on Thursday he was "deeply moved" as a father by shocking pictures of a three-year-old Syrian boy Aylan Kurdi's body lying on a Turkish beach after he, his brother and their mother drowned in an attempt to reach Europe by boat.
Aylan's father Abdullah, describing how the vessel they were travelling in overturned amid high waves, spoke of his devastation at the loss of "the most beautiful children in the world".
He said: "All I want is to be with my children at the moment."
The PM - who is today visiting Spain and Portugal for talks on his renegotiation of Britain's EU membership - said Britain's response to the crisis was "under review" and promised the UK would fulfil its "moral responsibilities".
As sources highlighted Mr Cameron's comment that "we are taking thousands of Syrian refugees and we will continue to do that", speculation mounted that the UK could be on the brink of agreeing to take in a four-figure number of vulnerable individuals from UN camps on the Syrian borders, rather than opening its doors to the desperate migrants who have poured into Europe across the Mediterranean.
The Government has so far declined to join a UN scheme for resettling the most vulnerable refugees, instead setting up its own programme that has admitted around 216 over the past year. Some 5,000 Syrians have been granted asylum in the last four years.
Britain has refused to join a EU scheme to resettle migrants arriving in frontline countries like Italy, Greece and Hungary.
By contrast, Germany has accepted 35,000 vulnerable Syrians through the UN scheme, Canada more than 10,000, Australia 5,600 and Switzerland 3,500.
European Commission president Jean-Claude Juncker is next week expected to call for EU countries to resettle some 160,000 migrants around the continent.
The PM has come under intense pressure to change his stance, including from backbench Tory MPs and leaders of other EU states.
French president Francois Hollande delivered a thinly-veiled swipe at the UK by complaining that some countries were not "shouldering their moral obligations".
And European Council president Donald Tusk, seen as an ally of the PM, upped the pressure further by arguing that "all EU members" needed to take their share of refugees.
Development charity Oxfam welcomed indications of a shift in policy. Chief executive Mark Goldring said: "We hope the Government will clarify as soon as possible the number of refugees from Syria the UK will resettle and by when.
"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."
And former Chief Rabbi Lord Sacks told BBC2's Newsnight the UK should accept 10,000 refugees in "a very clear and conspicuous humanitarian gesture", which he compared to the Kindertransport programme to save Jewish children ahead of the Second World War.
Former home secretary David Blunkett suggested that Britain needed to take in 25,000 refugees over a six-month period in order "to be taken seriously by anybody" in the discussion over the migration crisis.
Labour leadership candidates taking part in a televised debate on Sky News said the Prime Minister had failed to respond adequately to mounting problems over the summer.
Jeremy Corbyn said that accepting 4,000 additional refugees "doesn't seem like enough", adding: "We have to hold out the hand of humanity and support and friendship ... Every European country should do its best. Germany has shown the way."
Shadow home secretary Yvette Cooper, who has suggested the UK could take 10,000 more refugees, said: "Other countries are doing their best. We have got to do more as well."
Liz Kendall said she was "ashamed" of the "heartless and powerless" approach of the Prime Minister, saying Britain should be taking "something in the tens of thousands."
Andy Burnham - who is calling on the Government to present a plan for debate in the House of Commons on Monday - said the PM should be entering immediate talks with his EU counterparts on a deal to tackle the migration problem.
Liberal Democrat leader Tim Farron said: "It is embarrassing for the country that the Government has had to be dragged kicking and screaming to do the compassionate thing.
"But we must wait to see the detail - there can be no half measures, it's high time the UK took a full role in helping desperate people."
The British Red Cross announced it has launched an emergency appeal in the face of "one of the biggest humanitarian disasters of our time".
Head of emergencies Ben Webster said: "People are arriving in Europe with very little – some only a family photo and the clothes they left in.
"We urgently need supplies such as food, water, nappies and hygiene kits as well as emergency medical treatment – your support can make a huge difference.
"Millions of people are being forced to risk their lives in a desperate search for safety as conflicts rage around the world."