GENEVA — The United Nations will soon have to start cutting off lifesaving aid to people fleeing the war in Syria because the exodus has far outstripped financial support from international donors, one agency at the center of the humanitarian relief effort warned on Friday.

“The needs are rising exponentially, and we are broke,” Marixie Mercado, a spokeswoman for Unicef, told reporters in Geneva. “Across the region, a lot of our operations are going to have to start scaling down unless we get money.”

The warning came as President Bashar al-Assad, in a rare interview with a foreign media outlet, said that if Syria broke up or came under the control of “terrorist forces,” this would immediately spill over into neighboring countries first, and that a domino effect would reach countries across the Middle East.

Mr. Assad, who typically refers to his armed opponents as terrorists, appeared to be trying to capitalize on growing concerns among countries that support the uprising, including the United States, about extremist groups among the rebel ranks and the potential that the conflict could destabilize the region.