"Our view, based on what the Sudanese told us in Washington, is that the policy has been reversed," Bolton said on Friday.
"We interpret it very simply - they popped off, we stood up and they backed down."
Sudan had earlier warned that any nation pledging UN troops for its Darfur region was committing a "hostile act" in a "prelude to an invasion".
Bolton said that Sudan's ambassador to Washington and its UN envoy had told the state department and the UN Security Council president on Friday of a change in Khartoum's policy.
Bolton said: "The government of Sudan has backed down and this threat against potential-troop contributing countries I take to be null and void."
But Abdalmahmood Abdalhaleen Mohamad, Sudan's ambassador to the UN, did not explicitly retract the letter.
"There is no hostility in the letter at all, the letter is a very clear reflection of our views," Mohamad said.
"Our objective in sending this letter is dialogue, not confrontation."
"We did not intimidate, we are a victim of intimidation. We cannot intimidate other states."
Sudan has repeatedly refused a UN peacekeeping force of up to 22,500 troops and police for Darfur, as authorised by the UN Security Council.
An under-financed African Union (AU) force of 7,000 troops and monitors in Darfur has failed to stop the violence that has driven 2.5 million people from their homes and left at least 200,000 dead since 2003.
Bolton said that a number of governments said on Thursday that they thought it was time the Security Council issued a new statement on Sudan.
Bolton's draft of a statement, circulated on Thursday, would "deplore" the Sudan mission's attempt "to intimidate potential troop contributing countries volunteering forces for a peacekeeping mission in Darfur".
But some council members had reservations as to the tone of the draft statement.
Adamantois Vassilakis, Greece's UN ambassador, said: "For me what is important is how we find a solution to save lives. That is the most important thing".
Kofi Annan, the UN secretary-general, said on Thursday that the region was "on the brink of a catastrophic situation" and that a desperately needed humanitarian operation may have to be drastically cut.
The UN is currently planning to reinforce the AU troops in Darfur with 100 extra personnel who will run communications and other equipment ahead of the planned arrival of the larger UN operation.