About half of the refugees are in the Chad border city of Koruk and "are feeling terrified," said UNHCR spokeswoman Helene Caux.
"The city is in a very volatile area and there are a lot of armed men there," Caux told The Associated Press. "Their villages (in Sudan) have been looted and burned, they are telling the UNHCR team. And they say their villages are encircled by militias."
Most of the refugees so far are men, she said. But the arrivals are telling UNHCR that "thousands of women and children are on their way" to Chad, she added.
So far, UNHCR officials have been unable to reach all the refugees around Koruk. An roughly equal number of Darfur refugees have gathered in the nearby Chad town of Figeira. UNHCR is hoping to move the refugees to different camps the agency runs in Chad, Caux said.
U.N. officials say the worsening situation in Darfur has been exacerbated by a recent rebel attack on the capital of neighboring Chad. Chad has accused Sudan of backing those rebels in a bid to prevent deployment of European peacekeepers in an area of Chad where some 400,000 refugees are living.
Sudan's Arab-dominated government has been accused of unleashing janjaweed forces to commit atrocities against Darfur's ethnic African communities in the fight with rebel groups. At least 200,000 people have been killed and 2.5 million displaced since the violence began five years ago.
Caux said refugees are reporting that their villages were attacked by men on horses and camels, a description similar to those provided of earlier incidents involving the janjaweed.
The Sudanese army said its attacks forced rebels to retreat into neighboring Chad, a provocative accusation at a time of escalating tension between the two countries. Both nations accuse each other of hosting hostile rebel groups, allegations that became even more sensitive after Chadian rebels attacked Chad's capital last weekend.
Darfur rebels have denied any of their fighters were in the towns attacked by the government Friday and said some 200 people were killed in the attacks by helicopter gunships and fixed-wing aircraft.
Caux said UNHCR was looking at way to assist people still trapped in the three towns bombed by Sudan.
"Thousands of households have been directly affected by the bombings and attacks," she said.