Many have been forced to move on several occasions ahead of shifting frontlines to avoid crossfire and stay out of shelling range, UN High Commissioner for Refugees (UNHCR) spokesman Ron Redmond told journalists.
Civilians were fleeing Mannar in the far northeast of the island and travelling further north into Poonagary and Karachchi as fighting intensified, he said.
"Supplies of food, shelter materials, water and sanitations equipment, and fuel for the transportation of civilians are running dangerously low," he warned.
"Efforts by humanitarian agencies to replenish the stocks are hindered by the strict restrictions on the transport of goods into the region," Redmond added.
Sri Lanka's defence ministry said Thursday that at least 23 Tamil Tiger rebels and two soldiers were killed in the latest fighting.
Government troops kept up attacks against the Liberation Tigers of Tamil Eelam (LTTE) in Mullaittivu, Weli Oya and Vavuniya, with 75 rebels and three soldiers also wounded during