Major Gen. Ray Odierno, who commands 4th Infantry Division troops in Tikrit, says his men had captured several of Saddam's former bodyguards in the past month.
"If he makes a mistake, we'll have him," Odierno said.
Defense Secretary Donald H. Rumsfeld was in Tikrit on Friday.
He then went to Mosul, in the far north, to visit the 101st Airborne Division, which has made steady progress rebuilding schools, restoring water service, and launching a city council in the city.
"I think we have the formula here for success," Rumsfeld said Friday. "A lot of things have been accomplished. It's not going to be a straight steady path. In the future there will be difficulties."
In other developments:
Lt. Gen. Ricardo Sanchez said he had enough soldiers to accomplish the mission given to him by Washington, but its scope did not include guarding Iraq's porous borders or its thousands of miles of highways.
He warned that international forces were needed to reinforce existing coalition forces in tackling looming security threats, such as Iranian fighters or possible conflict between the Iraq's Sunni and Shiite Muslims sects.
"If a militia or an internal conflict of some nature were to erupt…that would be an additional security challenge out there that I do not have sufficient forces for," Sanchez said.
In Washington, Gen. Richard Myers, chairman of the Joint Chiefs of Staff, also said more international troops were necessary in Iraq — not because the current force needs help with security but because there is a need to counter the impression among Iraqis that the occupation is strictly an American operation.
Rumsfeld, who arrived in Baghdad Thursday, said the key to solving Iraq's problems is building up Iraqi security forces — not adding more U.S. soldiers.
Rumsfeld estimated that other countries could provide "maybe another division" in Iraq, or about 10,000 troops. There are now about 140,000 U.S. troops and about an additional 20,500 from 29 other countries, including Britain.
However, Australian Prime Minister John Howard says his nation will not send peacekeepers to Iraq even if the Security Council supports a new multinational force to help U.S. troops there.
The new U.S. proposal to the Security Council has been depicted as a new effort to get multinational help. But Deputy Defense Secretary Paul Wolfowitz says the United States has wanted U.N. support since the fall of Baghdad to bring in more international troops.
The U.S. draft resolution offers the United Nations a greater role in governing Iraq, preparing for the elections and peacekeeping in Iraq, although U.S. commanders will retain control.
The five veto-wielding permanent council members — the United States, France, Britain, Russia and China — met Thursday afternoon to discuss the draft. An informal meeting of all 15 council ambassadors was scheduled Friday, and talks were also taking place in capitals.
Washington said it wanted a quick vote on the proposed resolution. But the other Security Council nations had widely differing reactions.
Council supporters of the war — Britain, Bulgaria and Spain — rallied behind the resolution. Russia signaled cautious support and China was receptive.
France, Germany and Syria criticized it, but even France and Germany called the U.S. proposal a basis for negotiations.
The U.S. draft resolution invites the U.S.-appointed Iraqi Governing Council to cooperate with the United Nations and U.S. officials in Baghdad to set up elections, but contains no timeframe of when this should happen.
The resolution also asks the U.N. representative in Iraq to facilitate a "national dialogue and consensus building" to promote the political transition. But it doesn't provide for an expanded U.N. role in the security or economic areas.