More than 40,000 Syrians have fled to Iraqi Kurdistan in little over a week to escape an escalation in fighting between Kurdish rebels and extremists.

More than 40,000 Syrians have fled to Iraqi Kurdistan in little over a week to escape an escalation in fighting between Kurdish rebels and extremists, the United Nations said Friday.

"Since Thursday last week, we believe that 42,000 Syrians have now swept across Syria's far northeast border into Iraq," said Adrian Edwards, spokesman for the UN high commissioner for refugees.

On August 15, Iraq's autonomous Kurdistan region opened a crossing on its Tigris River border with Syria, which previously had been controlled tightly amid fears of a spillover of conflict.

Consequently, there was a massive influx of refugees, most of them Syrian Kurds.

With temperatures hitting 45 degrees Celsius (113 Fahrenheit), aid agencies have been scrambling to bring supplies to those fleeing and have helped move them to communities away from the border zone.

The biggest influx was last weekend, and aid agencies said that Iraqi Kurdistan had initially imposed a daily limit of 3,000, but it was not if this remained in place.

"Clearly, the principle has to be that people who need a safe haven have access to it," Edwards told reporters.

He said it was tough to say how many more refugees might be on their way.

"The numbers have been coming down, and we expect that trend to continue. But it's obviously contingent on the situation across the border in northern Syria itself," he explained.

Around two million Syrians have fled their homeland since the violent unrest in the country erupted in 2011.

Most have found a haven in Lebanon, Jordan and Turkey, while UN figures put the number in Iraq at least 197,000, including the latest influx.

Syria was hit by the unrest since mid-March 2011, where the Syrian government accuses foreign actors of orchestrating the conflict by supporting the militant opposition groups with arms and money.