"This temporary relocation takes place inside Afghanistan and will be for a limited period of time," the UN Assistance Mission in Afghanistan (UNAMA) said in a statement.
It said it is pulling its staff from its office in the Kunduz following the attack on the centre this weekend by protesters angry over the burning of copies of the Koran by US troops.
Relocation of the international staff at its Kunduz office will be temporary. The UN's mission said it intends to put in place additional arrangements and measures to ensure the office can continue to operate in safety.
The UNAMA office in Kunduz will continue to deliver the critical programmes in the region for the people who need them the most, it said. "UNAMA wants to reiterate that the organization is standing by the people of Afghanistan and will continue to stand as long as the people of Afghanistan want them."
No UN staff members were hurt in Saturday’s attack.
However, there were casualties among the Afghan National Security Forces (ANSF), which responded to the incident.
Among the demonstrators that had gathered outside the mission’s compound, four were killed.
More than 30 people have been killed in various cities since last week’s incident in which soldiers at a US air base inadvertently improperly disposed of a large number of Islamic religious materials, including copies of the Quran.