At the same time, Mr. Bush yet again sidestepped a question on the role of his top political adviser, Karl Rove, in the matter.
"We have a serious ongoing investigation here and it's being played out in the press," Mr. Bush said at an East Room news conference with the visiting prime minister of India.
Mr. Bush's latest comments marked a change of language and emphasis from his past assertions that anyone involved in leaking the name of agent Valerie Plame would be fired.
Mr. Bush said in June 2004 that he would fire anyone in his administration shown to have leaked information that exposed the identity of Plame, the wife of former Ambassador Joseph Wilson, an outspoken critic of the president's Iraq policy.
On Monday, however, he added the qualifier that it would have to be shown that a crime was committed. Not all such disclosures necessarily rise to the level of crime, reports CBS News White House Correspondent Mark Knoller.
Mr. Bush spoke a day after Time magazine's Matthew Cooper said that a 2003 phone call with Rove was the first he heard about the wife of Bush administration critic Joseph Wilson apparently working for the CIA.
A tempest has swirled around the leak of the CIA agent's name, apparently by Bush administration officials, in July 2003.
Some Democrats have called for Rove, whose title is deputy chief of staff, to be fired. They have suggested that he violated a 1982 federal law that prohibits the deliberate exposure of the name of a CIA agent.
"It's best people wait until the investigation is complete before you jump to conclusions. I don't know all the facts. I want to know all the facts," Mr. Bush said. "I would like this to end as quickly as possible. If someone committed a crime, they will no longer work in my administration."
Cooper also revealed over the weekend that Vice President Dick Cheney's chief of staff Lewis "Scooter" Libby was another of his sources for the Plame story.
Until it refused to issue more denials last week, the White House had insisted for nearly two years that Libby and Rove had no connection to the leak of Plame's identity.
On Sunday, Bush administration spokesman David Almacy declined to comment about Libby, citing the independent counsel's ongoing investigation.
Cooper said the 2003 phone call with Rove was the first time he had heard anything about Wilson's wife.
He said he had a subsequent conversation about Wilson and his wife with Libby.
According to Cooper, "Libby replied, 'Yeah, I've heard that too' or words to that effect" when Cooper asked if Libby had heard anything about Wilson's wife sending her husband to the African nation of Niger to investigate the possible sale of uranium to Iraq for nuclear weapons.
Recounting a July 11, 2003, conversation with Rove, Cooper recalled that Rove told him, "I've already said too much" after revealing that the wife of administration critic Joseph Wilson apparently worked at the CIA.
Cooper speculated that Rove could have been "worried about being indiscreet, or it could have meant he was late for a meeting or something else."
"I don't know, but that signoff has been in my memory for two years," Cooper wrote.
Cooper also said there may have been other government officials who were sources for his article. Time posted "A War on Wilson?" on its Web site on July 17, 2003.
In an effort to quell a chorus of calls to fire Rove, Republicans said Sunday that he first learned about Plame's identity from the news media.
"The information exonerates and vindicates, it does not implicate" Rove, Republican Party chairman Ken Mehlman said on NBC's "Meet the Press." "Folks involved in this, frankly, owe Karl Rove an apology."
There were no takers.
The White House's assurance in 2003 that Rove was not involved in the leak of the CIA officer's identity "was a lie" and Rove's credibility "is in shreds," said John Podesta, who was chief of staff in the Clinton White House.
It is unclear whether a journalist first revealed the information to Rove, as Mehlman said.
A lawyer familiar with Rove's grand jury testimony said Rove learned about the CIA officer either from the media or from someone in government who said the information came from a journalist. The lawyer spoke on condition of anonymity because the federal investigation is continuing.
Appearing on CBS' Face the Nation, Wilson pointed out that his wife "was covered according to the CIA, and the CIA made the referral" to the Justice Department for a criminal investigation.
Wilson said friends and neighbors of the couple did not know that she worked for the CIA and that they understood her to be "an energy analyst, an energy consultant."
Wilson said his wife was outed in a White House vendetta against him, and that President Bush must fire his adviser.
It's a question of trust with the American people," Wilson told CBS. "I also believe that the president should fire Karl Rove, because I believe that using the West Wing of the White House to be engaged in a smear campaign is an outrageous abuse of power."