France and the US are continuing to push for military action over alleged chemical weapons use by Syrian forces.
There are suggestions that President Barack Obama may be planning much wider action than the limited strikes that have been publicly proposed.
The reports emerged as senior US politicians were set to speak before a congressional committee, to rally support for intervention.
The UNHCR said in a statement on Tuesday: “Syria is haemorrhaging women, children and men who cross borders often with little more than the clothes on their backs.”
Around half of those forced to leave are children, UN agencies estimate, with about three-quarters of them under 11.
Just 118,000 refugee children have been able to continue in some sort of education, and only one-fifth have received some sort of counselling, with agencies warning of a “lost generation” of child refugees ill-equipped to help rebuild Syria in the future.
Lebanon has received the highest number of refugees, even though it is the smallest of Syria’s neighbours and one of the least able to cope.
There is now thought to be one Syrian refugee in Lebanon to roughly every six Lebanese. Jordan and Turkey have taken in the second and third highest numbers respectively.
As well as those who have left the country, a further 4.25 million have been displaced within Syria, the UNHCR says, meaning that more Syrians are now forcibly displaced than is the case with any other country.
Pointing out that more than 97% of Syria’s refugees are being hosted by countries in the surrounding region, the UNHCR said the influx was “placing an overwhelming burden on their infrastructures, economies and societies”.
It once again appealed for “massive international support” to help neighbouring countries deal with the crisis.
International aid agencies are also struggling to cope with the crisis, having only 47% of funds required to meet “basic refugee needs”, the UNHCR says.
“It took two years to reach the first million refugees. It took six months for the second million to be reached,” the UN High Commissioner for Refugees, Antonio Guterres, told the BBC.
He said officials could envisage three million refugees by the end of 2013.
One of the biggest single waves of refugees occurred in mid-August, when thousands from north-eastern Syria poured over the border into Iraqi Kurdistan.
Iraq has the fourth largest population of Syrian refugees, with over 170,000.
The UN says the conflict in Syria has caused the world’s worst refugee crisis for 20 years, with numbers not seen since the 1994 genocide in Rwanda.
More than 100,000 people are thought to have died since the uprising against Syrian President Bashar al-Assad began in March 2011.
The US and French governments are pushing for a military strike in reaction to what they say was a chemical weapons attack carried out by Syrian government forces on the outskirts of Damascus on 21 August.
The US has put the death toll at 1,429, including 426 children.
US Defence Secretary Chuck Hagel, Secretary of State John Kerry and Joint Chiefs of Staff Chairman Gen Martin Dempsey are due to appear later before the Senate Foreign Relations Committee.
They are trying to muster support for military intervention in Syria in the run-up to a vote in Congress expected next week.
On Monday a senior state department official told a gathering of Democrat politicians that a vote against military action would be America’s “Munich moment”, referring to the 1938 Munich Agreement which was seen as British and French appeasement of Nazi Germany before World War II.
The Obama administration’s document seeking Congress authorisation to conduct military strikes speaks of sending “a clear signal of American resolve”, aiming “to deter, disrupt, prevent and degrade the potential for future uses of chemical weapons or other weapons of mass destruction”.
Many analysts had assumed the strikes would be fairly limited in scope. But this may not be the case, according to Gen Jack Kean, a retired US Army vice-chief of staff who spoke to key Republican senators after they had been briefed at the White House on Monday.
He told the BBC that the senators had been “encouraged by what the president has told them in terms of the scale of robustness of what he intends to do” – and that they had been given the indication that strikes would go far beyond targeting the regime’s alleged chemical weapons.
“I think two things are central to what’s going to happen: he’s going to deter and degrade – and the important word is degrade – significant military capability of [Syrian President] Assad’s regime,” Gen Kean told the BBC.
Mr Obama already appears to have won the support of two of his fiercest foreign policy critics, Republican Senators John McCain and Lindsey Graham.
Mr McCain told reporters on Monday that a congressional vote against strike action “would be catastrophic in its consequences” for US credibility abroad.
French leaders, like their US counterparts, are pushing for military action.
A report presented to the French parliament by Prime Minister Jean-Marc Ayrault on Monday said the 21 August attack “could not have been ordered and carried out by anyone but the Syrian government”.
The report said the assault involved the “massive use of chemical agents”.
In an interview with French newspaper Le Figaro, Syrian President Bashar al-Assad denied being behind the attack, saying it would have been “illogical”.
He warned that foreign military action could ignite a wider regional conflict.
Nato Secretary General Anders Fogh Rasmussen also said on Monday that he was personally convinced a chemical attack took place and that the Assad government was responsible.
While saying he did not expect Nato to be involved in any action, he said there must be “a firm international response” to deter any future use of such weapons or else it would send a “dangerous signal to dictators all over the world”.
Syria is known to have extensive supplies of chemical weapons.