GENEVA — As fighting continues in Syria, an ‘‘extraordinary acceleration’’ in movements of refugees is compounding the difficulties for humanitarian relief efforts, the UN refugee agency said Tuesday.

Around 2,000 Syrians are crossing daily into Jordan amid continuing air and artillery attacks on towns near the southern border, said Adrian Edwards, a spokesman for the agency.

Thousands more Syrians are reported to be moving south from village to village seeking safety from the fighting before crossing into Jordan, he said.

The exodus has pushed the number of Syrian refugees to more than a quarter of a million, Edwards said. Of the total, Jordan now has more than 85,000 refugees and Turkey more than 78,000, the agency said, counting those who have registered or are awaiting registration with the agency. But many more refugees have not registered, and both countries say they have received roughly double the United Nations’ numbers.

The agency said more than 10,000 are waiting to cross into Turkey. The latest estimates come as Antonio Guterres, the UN’s high commissioner for refugees, and actress Angelina Jolie, a special envoy for the agency, toured the Za’atri camp in northern Jordan to draw attention to the needs of the swelling refugee population.

The camp, which opened on a windswept patch of desert close to the Syrian border at the end of July, already has 28,000 refugees, Edwards said.

Faced with a relentless flood of people fleeing the fighting, UN officials are talking to Jordanian authorities about finding new locations in less-harsh surroundings for facilities to receive them. Providing another snapshot of deteriorating living conditions inside Syria, the World Health Organization said that a UN mission to Homs last week found that more than half a million people needed aid, including health care, food, and water. The mission found the biggest hospital in Homs had been destroyed and only six of the 12 public hospitals, and eight of the 32 private hospitals, were still functional, although with severely reduced capacity.

At least half the doctors have left Homs, and only three surgeons now remain for an area with a population of more than 2 million. Many health facilities are staffed by volunteers without medical or health training and face critical shortages of medicines, WHO spokesman Tarik Jasarevic said.