More than 4,000 Iraqis from the northern city of Mosul have fled to Syria since the beginning of May, the UN has said, adding it is expecting up to 50,000 people to leave the ISIL-held city and cross the border.

Driving the refugee exodus appear to be reports that the Islamic State of Iraq and the Levant (ISIL, also known as ISIS) group has stepped up executions of men and boys in Fallujah - the other major Iraqi city still in their hands - since pro-government forces launched an offensive to re-take the city, the UN's refugee agency said on Friday.

According to UNHCR, a total of 4,266 people arrived this month at the al-Hol camp, 14km from the Iraqi border in Syria's northeastern Hasakah province.

"We've seen actually a spike in the numbers of Iraqi refugees who are risking the dangerous crossing into Syria in a desperate bid. Just picture this, we have refugees fleeing to Syria," UNHCR spokesperson Melissa Fleming said.

"The reasons for that are the pending battle to re-take it [Mosul]. They, I'm sure, hear what's going on in Fallujah and want to leave before they too are trapped. But also there is fighting in the surrounding areas that is driving people to leave."

The Iraqi army launched an offensive on Monday to dislodge ISIL from Fallujah, 50km west of Baghdad.

Fallujah was the first Iraqi city to fall under ISIL control in January 2014, and has been under a tight siege for about six months.

An estimated 50,000 people are trapped inside the city. ISIL fighters have prevented civilians from fleeing, but the Iraqi army said on Friday it had managed to evacuate some families..

"Our forces evacuated 460 people... most of them women and children," said police Lieutenant General Raed Shakir Jawdat.

However, the Norwegian Refugee Council said that of the families it knew of that had escaped, the vast majority were from the outskirts of the city.

"The situation inside Fallujah is getting critical by the day," said Nasr Muflahi, NRC's Iraq director.

"We are now hearing reports of contaminated water being used for drinking, while entire neighbourhoods are being displaced within the battle zone with no safe way out."

INSIDE STORY: Retaking Fallujah

Iraqi forces, with help from a US-led coalition, are expected to push later this year to retake Mosul, which has been held by ISIL since June 2014.

The refugees from Mosul, who walked for several days before reaching the al-Hol camp, "are living now in relative safety - if you can say that for Syria," Fleming said.

"Right now it's a bit over 4,000 but it is in anticipation of 50,000. There are contingencies for potential numbers who could be coming in ... They don't have many other options of places to flee in that region, so we're getting ready."