* UN refugee agency says "asking Europe to do more"
* Nearly 3 mln Syrians in Lebanon, Jordan, Turkey, Iraq (Adds more figures from UNHCR)
By Tom Miles
GENEVA, July 11 (Reuters) - Europe must open its doors to more Syrian refugees, having welcomed only a "miniscule" number while Syria's neighbours have reached "saturation point", the U.N. refugee agency UNHCR said on Friday.
Syrian refugees, whose numbers are set to pass 3 million in the next few weeks, are almost all in Lebanon, Jordan and Turkey, with smaller numbers in Iraq and Egypt. But that is changing, said UNHCR chief spokeswoman Melissa Fleming.
"The trend is obvious now, they're moving beyond the neighbouring countries. The neighbouring countries have reached saturation point. And many Syrians are now seeking refuge in Europe and we're asking Europe to do more."
In more than 3 years of war, very few Syrians have made it to continental Europe. There have been 123,600 asylum requests, mostly in Sweden and Germany, but that figure includes double counting, since some have asked for asylum in several countries.
"Relative to the 2.9 million refugees in the countries immediately surrounding Syria, these numbers are small, in fact they're miniscule. They represent only 4 percent of Syrian refugees," Fleming told a news briefing in Geneva.
"Just to put this into perspective, Europe has a population of 670 million people. Contrast that to Lebanon, which has a population of 4.4 million people and has received 1.1 million refugees."
Syria could potentially produce millions more refugees, since more than 6.5 million people are displaced within the country and there is no end in sight to the war.
Although the European Union (EU) has been a leading donor for the humanitarian aid effort in Syria, some countries on the continent have "terrible" conditions for receiving refugees. Some detain them, and some - Bulgaria, Spain and Greece - have tried to reject them before they land in unseaworthy boats.
"There was one case in which survivors claim their boat was being turned around by an official boat and in the process of being turned around it capsized and many of the passengers died. This is unacceptable," Fleming said.
A UNCHR report said that happened near the Greek coast of Farmakonisi on Jan. 20 this year, and 12 women and children from Syria and Afghanistan died.
One country in Europe - Russia - has sent 12 refugees back to Syria, breaking an international law principle that no would-be refugees should be sent back to a place where their life or liberty would be at risk, the report said.
Russia - an ally of President Bashar al-Assad - has not granted refugee status to a single Syrian, although it granted 1,193 temporary asylum requests in 2013, it added.
So far 17 European countries have offered to resettle 31,800 of the most needy refugees, but UNHCR wants the continent to host 100,000 such cases, an average of about 3,000 per country.
Fleming told Reuters that European countries should go further and admit everyone who needed international protection and who wanted to come, without specifying a number.
She said most refugees would want to remain closer to home in an Arabic-speaking community.
"Some people, when we offer to resettle them, have turned it down."
She blamed tabloid newspapers for "awful and distorted headlines claiming that floods of Syrians were about to arrive".
"We're asking for other forms of protection which include family reunification, extending student visas, offering scholarships, all kind of different mechanisms that could allow for the admission of more Syrians." (Reporting by Tom Miles; Editing by Stephanie Nebehay and Ralph Boulton)
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