WASHINGTON -- President Bush questioned on Tuesday whether investigators would be able to determine who leaked the identity of an undercover CIA officer but said his staff was cooperating. "I want to know the truth," he said.

Bush's chief of staff, Andrew Card, urged some 2,000 White House employees to turn over any relevant documents by Tuesday night. White House lawyers will screen the materials and decide which ones to send to the Justice Department as part of a criminal inquiry into the leak, Bush spokesman Scott McClellan said.

McClellan said it could take as long as two weeks to check those submissions for relevance.

The spokesman would not rule out the possibility that the White House would invoke executive privilege to shield sensitive documents from the Justice Department's inquiry. He said it was premature to talk about such a step.

Bush renewed his pledge to cooperate with the investigation to "come to the bottom of this."

But he said success was not guaranteed, and he turned reporters' questions back on them at the end of a Cabinet meeting.

"You tell me: How many sources have you had that's leaked information, that you've exposed or had been exposed? Probably none," he said.

"This is a large administration and there's a lot of senior officials," Bush told journalists. "I have no idea whether we'll find out who the leaker is, partially because, in all due respect to your profession, you do a very good job of protecting the leakers."

Adding a note of optimism, Bush also said, "But we'll find out."

Investigators are trying to determine who leaked to columnist Robert Novak and two Newsday journalists the identity of Valerie Plame, a CIA operations officer who has served overseas. She is married to former Ambassador Joseph Wilson, who publicly accused the Bush administration of manipulating intelligence to exaggerate the threat from Iraq.

The White House will not send investigators material deemed irrelevant, McClellan said.

"You don't want to overburden the Department of Justice with documents that have no relevance or are not responsive to their request," he said. "They're welcome to look at the other documents -- that's not an issue -- that are not responsive to their request."

But a Democratic lawmaker questioned White House intentions.

"I am very troubled by the fact that the White House counsel seems to be a gatekeeper, and I want to know what precautions Justice is taking to ensure that it gets all relevant information from the administration," said Sen. Charles Schumer, D-N.Y.

The Justice Department wants the White House to turn over all materials by mid-October, and the more relevant ones sooner.

The White House had sought to collect the last of its staff members' documents by a self-imposed deadline of 5 p.m. EDT Tuesday but acknowledged not all would be in.

In a memo Tuesday afternoon, Card told his staff, "The sooner we complete the search and delivery of documents, the sooner the Justice Department can complete its inquiry -- and the sooner we can all return our full attention to doing the work of the people that the president has entrusted to us."

Officials at the Defense and State departments and the CIA also have been asked to preserve any relevant documents. But there was no indication that they have been ordered to turn over such documents to the Justice Department.

McClellan firmly ruled out any role by three administration officials in the leak: political adviser Karl Rove, Vice President Dick Cheney's chief of staff, I. Lewis "Scooter" Libby, and National Security Council official Elliott Abrams.

But Rep. John Conyers, D-Mich., the top Democrat on the House Judiciary Committee, said Rove should resign.

"In my view, it is shameful and unethical that an administration that promised to govern with 'honor and integrity' and 'change the tone' in Washington has now engaged in an orchestrated campaign to smear and intimidate truth-telling critics, placing them in possible physical harm and impairing the efforts and operations of the CIA," Conyers said.