The leader of the embattled Taliban, Mullah Mohammad Omar, made the threat in a telephone interview with the British Broadcasting Corp.
Omar dismissed Taliban military reverses, and insisted that "the current situation in Afghanistan is related to a bigger cause, that is the destruction of America."
Asked whether he had a "concrete plan" to implement the destruction of the United States, Omar responded, "The plan is going ahead and, God willing, it is being implemented. But it is a huge task, which is beyond the will and comprehension of human beings.
"If God's help is with us, this will happen within a short period of time," Omar added, according to the BBC. "Keep in mind this prediction."
However, when asked whether his threat and "plan" were related to claims by Osama bin Laden to have nuclear and biological weapons, Omar said: "This is not a matter of weapons."
"We are hopeful for God's help" the BBC quoted him as saying. "The real matter is the extinction of America. And, God willing, it will fall to the ground."
In Washington, the senior U.S. official, who is familiar with national security concerns, was both dismissive and cautious.
"We do not give his threat too much credibility," the official said. "However, at this time we are taking all threats" -- regardless of credibility -- "very, very seriously."
The threat comes against a backdrop of heightened concern over attempts by bin Laden -- who has been harbored by the Taliban -- to acquire weapons of mass destruction. Bin Laden said last week he has nuclear and chemical weapons that he would use as a deterrent, though U.S. officials say they have no reason to believe that he has acquired nuclear weapons.
Thursday, Homeland Security Director Tom Ridge confirmed that instructions on making nuclear devices had been found at a "safe house" in Kabul that had been hastily abandoned by members of bin Laden's al Qaida network, which the United States blames for the terrorist attacks in New York and on the Pentagon Sept. 11. Ridge said the materials could have been downloaded from the Internet. But he said it underscores the need to be prepared for such threats.
Meanwhile, the Justice Department is continuing to seek interviews with more than 5,000 immigrant men between the ages 18 and 33 who have entered the United States since Jan. 1, 2000, from countries considered likely venues for terrorist plotting.
The department has said the men are not terrorist suspects, but may have information relating to others who are.
The interviews are being carried out, in large part, by state and local law enforcement at the request of the 94 U.S. attorneys offices across the country.