Up to 14 people were killed, including one U.S. and one South Korean soldier, in the Bagram air base attack which rebels said was aimed at Cheney.
He had been in his room at the base where he had unexpectedly had to stay the night after bad weather forced postponement of his trip to the capital, Kabul, about 60 km (40 miles) away.
"At 10 a.m. I heard a loud boom," Cheney said.
Base authorities sounded a red alert and secret service officials told Cheney there had been a suspected suicide attack.
"They moved me for a relatively brief period of time to one of the bomb shelters nearby," he said. "As the situation settled down and they got a better sense in terms of what was going on, then I went back to my room until it was time to leave."
NATO's death toll in the attack was four, officials said. A Reuters photographer at the scene saw an additional 10 bodies, putting the total at 14.
A U.S. government contractor, whose nationality was unknown, was among those killed and 27 people were wounded, NATO said.
"We wanted to target ... Cheney," Taliban spokesman Mullah Hayat Khan told Reuters by phone from an undisclosed location.
In Washington, the White House said it was unclear how the suicide bomber had been able to get close to the base.
"At this point, people are still investigating what happened, so we don't have a firm answer for you," White House spokesman Tony Snow told reporters.
Snow called it an "isolated attack" and declined to say whether it was a sign of Taliban strength.
"We've often said about acts of terror: An individual who wants to commit an act of violence or kill him or herself -- very difficult to stop," Snow said.
U.S.
President George W. Bush's initial reaction to the attack was concern about whether Cheney was all right, he said.
Soon after the blast, Cheney -- who officials say was never in danger from the blast at the sprawling base -- went ahead with talks with Afghan President Hamid Karzai in the capital.
Asked if he had ever considered changing his plans to go to Kabul, Cheney said that was "never an option."
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