In another effort to undo the legacy of George W. Bush's presidency, the Obama administration is searching for alternatives to the term "war on terror."

In recent days, Obama's national-security officials have had brainstorming sessions to come up with different ways to describe the U.S. government's efforts to defeat Al Qaeda and other terrorist groups in Afghanistan, Iraq and Pakistan, according to administration sources who asked not to be identified talking about private discussions.

What's being sought is a more precise phrase that can recast the U.S. government's counterterrorism fight in ideological as well as military terms. Obama publicly signaled the new approach this week. When asked about the "war on terror" phrase by CNN's Anderson Cooper, Obama said, "Well you know, I think it is very important for us to recognize that we have a battle or a war against some terrorist organizations … Words matter in this situation because one of the ways we're going to win this struggle is through the battle of hearts and minds."

Although partly symbolic, the search for new terror terminology reflects an internal government debate that predates the new administration. Critics have long decried the use of the phrase "war on terror" on the grounds that terrorism is a tactic, not an identifiable enemy. Years ago, State and Defense Department officials tried to move away from the phrase "war on terror," proposing instead to call it a "Struggle Against Violent Extremism," or SAVE.

But when word of the suggested change leaked to the media, President Bush displayed his annoyance at the idea during meeting of National Security Council officials. "The president unleashed over this," said one participant in the meeting who asked not to be identified talking about an internal discussion. "He made it perfectly clear that the American public understood what the war on terror was. … He was clearly irritated about this. That put an end to it."

Indeed, the "war on terror" was one of the signature phrases of the Bush presidency. It was formally declared in Bush's nationally televised speech to Congress on Sept. 20, 2001—his first after the 9/11 terror attacks—when the president cast the government's response in such sweeping terms that, according to critics, it ultimately opened the door for the invasion of Iraq. "Our enemy is a radical network of terrorists and every government that supports them," Bush said then. "Our war on terror begins with Al Qaeda, but it does not end there. It will not end until every terrorist group of global reach has been found, stopped and defeated."