Bush has vowed to go after bin Laden but has given few details of how he plans to go about it. US officials have declined to give updated figures on military deployments around Afghanistan.
Iran, which already has more than two million Afghan refugees, has sealed the border to keep out more refugees. Tehran has agreed to help refugee agencies, but wants any new camps to be set up near the border inside Afghanistan.
Grandi said Iran had agreed to allow the UNHCR to set up bases in the provinces of Khorasan and Sistan-Baluchestan to send consignments and staff to refugee camps.
"We prefer the camps to be on the border because refugees should feel safe in their camps and we should be able to reach them without any problem. We are negotiating on the location," said Grandi, whose mission has been expelled from Afghanistan.
The UNHCR had asked Iran to admit 400,000 refugees for six months, he said.
The agency was hoping to raise $US60 million ($148 million) to finance the project.
After decades of war and three years of severe drought, about five million Afghans are already dependent on outside aid for survival, according to the UN.
"We hope the Afghan people will not be forgotten. Some three million people were at risk at the time we were forced to leave Afghanistan," Grandi said.
"They might have only one solution: to cross the borders and seek refuge in neighbouring countries."
Grandi said there was no big number of refugees massing at any border so far.
"We have little information due to the cut in communication links, but we know a lot of people are moving in search of food. They are afraid of foreign military action and of the internal strife," he said.
"Right now people are moving to the countryside, but we have to understand there will be a time when waves of refugees will emerge.
"We have to find ways to get the world's attention to this sensitive issue."
- REUTERS
Map: Opposing forces in the war against terror
Afghanistan facts and links
Full coverage: Terror in America