The terror group also welcomed the U.S. Republican electoral defeat that led to the departure of Defense Secretary Donald H. Rumsfeld and vowed to continue its fight until the White House is blown up.
"The al Qaeda army has 12,000 fighters in Iraq, and they have vowed to die for God's sake," a man who identified himself as Abu Hamza al-Muhajir said in an audio tape released Friday. He also claimed to have another 10,000 unequipped fighters ready to go into battle.
In the tape, al-Muhajir praised the outcome of Tuesday's elections in which Democrats swept to power in the House and the Senate, in large part due to U.S. voter dissatisfaction over the handling of the war in Iraq.
"The American people have put their feet on the right path by ... realizing their president's betrayal in supporting Israel," the terror leader said. "So they voted for something reasonable in the last elections." He did not explain his logic.
"The change in leadership will not have a direct impact on what we could or don't do in Iraq," U.S. Joint Chiefs chairman Gen. Peter Pace said Friday on CBS News' The Early Show.
The authenticity of the 22-minute tape and al-Muhajir's identity could not be verified, but it appeared on a Web site commonly used by Islamic militants.
The CIA said it was analyzing the tape but declined to comment further.
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Al-Muhajir, also known as Abu Ayyub al-Masri, took over as leader of al Qaeda in Iraq, one of the country's deadliest terror groups, after his predecessor, Abu Musab al-Zarqawi, was killed in a U.S. air strike north of Baghdad in June.
The terrorist chieftain called Bush "the most stupid president" in U.S. history and declared that was allowing al Qaeda to move to victory more quickly than expected.
But he challenged Mr. Bush not to leave Iraq, saying: "We haven't had enough of your blood yet."
"We call on the lame duck (President Bush) not to hurry his escape the way the defense secretary did," al-Muhajir said in reference to Rumsfeld's resignation as Pentagon chief on Wednesday.
"Remain steadfast on the battlefield, you coward," al-Muhajir said to the American leader.
The al Qaeda commander also promised that his militants would not give up their fight until they had blown up the White House.
"We will not rest from our Jihad (holy war) until we are under the olive trees of Rumieh and we have blown up the filthiest house — which is called the White House," al-Muhajir said.
His reference to the "olive trees of Rumieh" was not clear, but Diaa Rashwan, an Egyptian terrorism specialist, said he appeared to be referring to ancient biblical Palestine.
Ben Venzke, head of the Virginia-based IntelCenter, which monitors terrorism communications, said al Qaeda in Iraq was swift to respond to events in the United States.
"It shows a remarkable degree of speed in being responsive," he said.
Rashwan said Rumsfeld's resignation and the Democrats winning both houses of Congress will have a "moral impact" on U.S. troops in Iraq.
"Al-Muhajir realizes this, and that's why he asked Bush to keep his forces and fight," Rashwan said.
But the Egyptian terrorism expert also said al-Muhajir's claim to have 12,000 fighters was "propaganda."
"This number is very big, and it is an extreme exaggeration, because U.S. reports state that most of al Qaeda fighters are non-Iraqis," he said. "If there are all those non-Iraqi fighters, where would they be hiding?"
In the audio tape, al-Muhajir also called on Sunni Muslims to pledge their allegiance to a new state that militants have said they created in Iraq, stating its ruler was Abu Omar al-Baghdadi.
"I vow allegiance to you," he said, addressing al-Baghdadi as the "ruler of believers" and placing al Qaeda in Iraq fighters under his command.
In October, an Iraqi militant umbrella group that includes al Qaeda in Iraq announced that it had established an Iraqi Islamic State, consisting of several Sunni-dominated provinces including Baghdad.
Iraqi authorities have dismissed their claim.
Venzke said al-Muhajir's claim of allegiance to al-Baghdadi shows that there is increased backing among Sunni insurgents for the creation of an Islamic state in Iraq.
"It shows a growing sort of support and organizational presence around this idea," he said.
Iraqi security officials said last month they were close to capturing or killing al-Muhajir and released a captured video showing the terror chief teaching followers how to build a car bomb. The video was the first to show the militant leader's face, though U.S. and Iraqi military officials have shown photos of him.
Al Qaeda in Iraq last released an audio tape purportedly from al-Muhajir in late September, where he called for nuclear scientists to join his group's holy war and urged insurgents to kidnap Westerners.