"We ought to get to the bottom of it so it can be evaluated, again, by the American people," Senate Judiciary Committee Chairman Arlen Specter said Sunday.
In a federal court filing last week, the prosecutor in the case said Cheney's former chief of staff, I. Lewis Libby, testified before a grand jury that he was authorized by Mr. Bush, through Cheney, to leak information from a classified document that detailed intelligence agencies' conclusions about weapons of mass destruction in Iraq.
A lawyer knowledgeable about the case said Saturday that Mr. Bush declassified sensitive intelligence in 2003 and authorized its public disclosure to rebut Iraq war critics, but he did not specifically direct that Libby be the one to disseminate the information.
Specter, R-Pa., did not dispute the White House claim that the president has the power to declassify secrets, reports CBS News correspondent Peter Maer.
"It may turn out that he had the authority to make the disclosures that were made but it was not the right way to go about it," Specter said.
But, he added, "it was not the right way to go about it because we ought not to have leaks in government."
He called for a detailed explanation of the leaks.
"I think that it is necessary for the president and vice president to tell the American people exactly what happened," Specter told "Fox News Sunday."
"I do say that there's been enough of a showing here with what's been filed of record in court that the president of the United States owes a specific explanation to the American people ... about exactly what he did," Specter said.
Libby faces trial, likely in January, on charges of perjury and obstruction of justice for allegedly lying to the grand jury and investigators about what he told reporters about CIA officer Valerie Plame.
Special Counsel Patrick Fitzgerald did not say in the filing that Cheney authorized Libby to leak Plame's identity, and Mr. Bush is not accused of doing anything illegal.
Sen. John Kerry, the Massachusetts Democrat who ran against Mr. Bush for president in 2004, said it was wrong for Mr. Bush to declassify information selectively "in order to buttress phony arguments to go to war " and to attack people politically.
"This was not a declassification in order to really educate America. This was a declassification in order to mislead America," Kerry said on "Meet the Press" on NBC. "I think it's a disgrace."
Wilson said Sunday that Mr. Bush and Cheney should release transcripts of their interviews with Fitzgerald.
"It seems to me that first and foremost, the White House needs to come clean on this matter," Wilson said on ABC's "This Week." "My own view of this is that the White House owes the American people and particularly our service people who have been sent into war, an apology for having misrepresented the facts."
The lawyer knowledgeable about the case said Mr. Bush instructed Cheney to "get it out" and left the details about disseminating the intelligence to him. The lawyer, who spoke on condition of anonymity because of the sensitivity of the case for the White House, said Cheney chose Libby and communicated the president's wishes to his then-top aide.
"I don't think there's any evidence that the president told the vice president to go leak information to the press," said Sen. John Kyl, R-Ariz.
Kyl said on CNN's "Late Edition" that a better way for the administration to have tried to counter Wilson's claims in a New York Times op-ed would have been to "have all of the press be given" the declassified intelligence material.
It is not known when the conversation between Mr. Bush and Cheney took place. The White House has declined to provide the date when the president used his authority to declassify the portions of the October 2002 National Intelligence Estimate.
"There has to be a detailed explanation as to precisely what Vice President Cheney did, what the president said to him and an explanation by the president as to what he said," Specter said.