The huge influx of migrants from Syria and Iraq is putting the future of the European Union in "grave danger", French prime minister Manuel Valls has warned.
And former prime minister Tony Blair said that the EU would face "a huge political problem" if terrorists were shown to be entering Europe among the flow of refugees. The EU needs to pool military capabilities more effectively in response to the migration crisis, he said.
Prime Minister David Cameron acknowledged that Europe was experiencing "very great" pressure from migration and said the UK was "happy to do more" to strengthen external EU border controls, even though it is not part of the Schengen border-free zone which covers most of the continent.
Mr Valls said that European societies could be "totally destabilised" unless the EU imposes tighter controls at external borders.
The French PM told the BBC: "It's Europe that could die, not the Schengen area. If Europe can't protect its own borders, it's the very idea of Europe that could be thrown into doubt.
"It could disappear, of course - the European project, not Europe itself, not our values, but the concept we have of Europe, that the founding fathers had of Europe.
"Yes, that is in very grave danger. That's why you need border guards, border controls on the external borders of the European Union."
He left little doubt that he believes German chancellor Angela Merkel's announcement last year that her country would welcome thousands of refugees had encouraged more people to come to Europe.
"We need to help Germany," said the French PM. "But the main message we must send now with the greatest of firmness is to say that we will not take in all the refugees in Europe.
"A message that says 'Come, you will be welcome' provokes major shifts of population. If you say anything in Europe today, a few seconds later it is on the smartphones of people in refugee camps near Libya."
Responding to Mr Valls's comments during a visit to Czech capital Prague, Mr Cameron said: "We are not in Schengen, so let's be clear Britain has its own border controls and we are going to maintain those border controls whatever anyone else in Europe decides to do.
"Since I've been Prime Minister, we've prevented 96,000 people coming into our country, including well over 6,000 EU citizens. We have borders, we use our borders and we will continue to use our borders.
"What I would say to European colleagues and Schengen members is that you have to have a strong external border if you are not going to have internal borders. Britain recognises that, even outside Schengen, clearly it's in our interests that we have this situation brought rapidly under control.
"That's why we already make contributions to the external border that Europe has and we are happy to do more - not just in terms of controlling numbers, but also the systems that we are members of (which) make sure we know about criminals crossing borders, so we know before they even get to Britain, make sure we know about terrorists crossing borders, so we know that even before they get to Britain."
Asked if the EU could survive the migration crisis, Mr Blair told Fox Business: "It can survive it, but it's going to be a big challenge. And if you did get terrorist activity coming in on the back of it, it would be a huge political problem."
The former PM said there was "a genuine desire to be generous" to refugees, but Europe needed to respond to public concern about security.
"There's an issue of numbers - how many people can Europe absorb? - but there's also an issue of security - can we be sure exactly who is coming in?" he said. "Those two issues combined make it a very, very tough political challenge."
Mr Blair said the EU needs to "up its game in defence terms" in response to the turmoil in Syria and potentially in Libya.
"What I think is important is that Europe pools its force capabilities in a more effective way," he said. "I don't mean that we amalgamate all the armed forces in one big EU army - that would never work. But how we make sure, particularly when we are working alongside the US, that Europe is stepping up with its own responsibilities, that's the challenge."
Responding to Mr Valls's comments, Liberal Democrat leader Tim Farron said: "The refugee crisis is putting huge strain on countries across Europe, but we need to come together and solve problems. Pulling up the drawbridge is not the solution.
"This might be Europe's biggest challenge yet but it is one we need to face together by creating safe and legal routes for refugees."
Newly updated figures revealed that the number of people who applied for asylum in the EU last year has passed 1.25 million.
Data compiled by the EU's statistical office Eurostat showed that 1,253,820 main applicants and dependants lodged claims in the 28 member states.
Last month the total for 2015 officially passed one million. The final tally will be higher as several countries have not yet provided figures for November and December.
Analysis of the figures revealed earlier this month that the number of people seeking asylum in the UK passed 5,000 in a month for the first time in at least six years.