The president, returning to the White House from Camp David, called on the government in Damascus to work with the U.S. and British officials.
"The Syrian government needs to cooperate with the United States and our coalition partners and not harbor any Baathists, any military officials, any people who need to be held to account for their tenure," Mr. Bush said.
When asked if the U.S. would intervene militarily if Syria did not hand over Iraqi leaders, the president said, "They just need to cooperate."
The president's comments came after U.S. Secretary of Defense told NBC's "Meet the Press" Sunday there is "no question" that some senior Iraqi officials fled Iraq for or through Syria. Some have sought refuge there, he said, while others have continued on to other regions.
Rumsfeld did not specify which Iraqi leaders he was talking about. He also refused to comment on whether the United States was prepared to take action against Syria for harboring Iraqi fugitives.
"We certainly are hopeful Syria will not become a haven for war criminals or terrorists," the defense secretary said.
Rumsfeld also said the majority of foreign fighters have faced in Baghdad over the past 24 hours have been Syrian nationals, including a gunman who shot and killed a Marine Saturday.
He told CBS' "Face the Nation" that U.S. soldiers encountered a bus full of Syrians who were carrying roughly $600,000 in cash, and flyers "suggesting that people would be rewarded for killing Americans."
Secretary of State Colin Powell told BBC 1's "Breakfast with Frost" American concerns over Syria's actions date back to before the war in Iraq.
"We have designated Syria for years as a state that sponsors terrorism, and we have discussed this with the Syrians on many occasions," the secretary said.
Powell also reiterated charges that Syria has allowed military and other banned equipment into Iraqi through their border.
"We are concerned that materials have flowed through Syria to the Iraqi regime over the years. We are making this point clearly and in a very direct manner to the Syrians," Powell said, adding he hoped Syrian leaders would "respond accordingly."
In Washington, Syrian deputy ambassador Imad Moustapha told "Meet the Press" that the allegations against his country are false, flatly denying his country is knowingly sheltering Iraqi leaders.
"The answer is no," he said. The ambassador called the accusations part of a campaign of misinformation about Syria, saying only humanitarian aid is crossing the Syria-Iraq border.
"This is an unfair campaign," Moustapha said. "We don't even have an ambassador to Iraq. How would we know what?s going on there?"