* Jordan and Turkey already host more than 100,000 each
* 358,000 Syrians registered in region, true number higher
* Europe must keep borders open to Syrians, UNHCR says
By Stephanie Nebehay
GENEVA, Oct 23 (Reuters) - Lebanon has become the third of Syria's neighbours after Turkey and Jordan to register more than 100,000 refugees from Syria's civil war, the U.N. refugee agency said on Tuesday.
In all, more than 358,000 Syrians fleeing the 19-month-old conflict have registered in four neighbouring states, including Iraq, and tens of thousands more have fled but not sought international assistance, it said.
Lebanon has seen its own Syrian-related unrest, exacerbated by the assassination last week of a senior intelligence officer who had pursued evidence of Syrian attempts to destabilise Lebanon.
"In Lebanon, we have not seen, despite the recent turmoil, a decrease in people crossing," UNHCR chief spokeswoman Melissa Fleming told a news briefing in Geneva.
Jordan now has 105,737 registered Syrian refugees, followed by 101,834 in Turkey, 101,283 in Lebanon and 42,661 in Iraq. The UNHCR has forecast that a total of up to 700,000 Syrian refugees may have fled abroad by the end of the year.
Fleming called for international support for countries hosting the refugees. "These nations should not be expected to carry the entire burden alone," she said.
Some 6,815 Syrian refugees have registered in North Africa, nearly all in Egypt, but the Egyptian government has said that there are up to 150,000 Syrians in the country, who may still seek refugee status if their money or visas run out, Fleming said.
She also urged European countries to keep open their borders for people fleeing Syria and grant them asylum.
Some 16,474 Syrians sought asylum in the European Union, Norway or Switzerland between January 2011 and August 2012, according to the UNHCR's latest figures.
"We have seen some uneven treatment depending on the asylum systems in the countries in Europe. However, we have not seen any indication that borders are closed," Fleming said.
International mediator Lakhdar Brahimi, who held talks in Damascus on Sunday with President Bashar al-Assad, has proposed that Assad's forces and the rebels hold fire during the three-day Islamic holiday of Eid al-Adha which starts on Friday.
The UNHCR said that "in anticipation of a possible truce during Eid", it was preparing 10,000 emergency relief kits for displaced families in areas around Aleppo, Syria's largest city, and Idlib that it had so far been unable to reach.
Syrian rebels cast doubt on Monday on prospects for a temporary truce, saying it was not clear how it could be implemented. (Reporting by Stephanie Nebehay; Editing by Kevin Liffey)