UN officials expect more than a million Afghan refugees to cross into Pakistan following the air strikes.
But President Pervez Musharraf said his country could not take another influx of refugees.
Reports from Kabul and other Afghan cities indicate there has been a massive civilian exodus.
News agencies in Pakistan quote UN officials as saying a million refugees will try to cross the border with Pakistan. Another 500,000 would try to enter other countries on Afghanistan's border, they said.
Yousaf Hassan, the UN High Commission for Refugees' spokesman, said the organisation asked Pakistan to open its borders for humanitarian reasons.
He said 1.5 million refugees were expected to cross into countries bordering Afghanistan but the "largest influx was expected into Pakistan".
World Food Programme workers said they expected to carrying on working inside Afghanistan where millions of people are facing starvation.
US Air Force C-17 transport planes have already dropped food rations intended for 37,500 displaced Afghans.