During a call to the current head of the European Union, Danish Prime Minister Anders Fogh Rasmussen, Bush made it clear he felt "not his responsibility to define" who or what would replace the Iraqi president, according to one diplomat.
Bush "expressed the view that any alternative is preferable" to Saddam, added the diplomat.
A second official from another European country agreed that Bush had "said it was up to the international community to help set up what follows" once the government in Baghdad had been toppled.
The news comes amid growing concern in the international community about the possible effects of a U.S. military strike to effect what Americans call regime change in the beleaguered Arab nation.
U.N. Secretary-General Kofi Annan, gave voice to these concerns Monday.
Asked about the consequences of a U.S. attack on Iraq, he said it would be "difficult to predict.
"I know many people are worried about unexpected consequences, and the question is -- the morning after," he said, explaining, "What sort of Iraq do we wake up to after the bombing, and what happens in the region? What impact could it have? These are questions leaders I have spoken to have posed."
The secretary-general said he did not want to "throw out any guesses, but I am concerned as well."
Annan advised the international community to wait until President Bush's address to the General Assembly later in the week.
Earlier, Annan was asked if any nation had come forward with an "initiative in the Security Council or by any other means before they take any further military action against Iraq?" and he replied, that he had spoken with Bush and British Prime Minister Tony Blair "and I would suggest we all wait to hear what President Bush has to say on Thursday" when he addresses the opening of the annual general debate in the U.N. General Assembly.
When asked about French President Jacques Chirac's proposal the council might serve an ultimatum on Baghdad to allow U.N. weapons inspectors, barred since late 1998 from Iraq, to return, Annan told reporters, "I think it is important to stress that the council, which has been seized with this Iraqi issue for so long, should have something to say. I think it is appropriate that the council pronounces itself on the issue."
Bush, who last week telephoned the leaders of key Security Council members -- France, Russia and China -- spoke Monday with Annan, Rasmussen, and Turkish President Acmet Necdet Sezer, spokesman Ari Fleischer said Monday morning.
Calls were scheduled for later in the day to George Robertson, NATO's secretary-general, Egyptian President Hosni Mubarak and Saudi Crown Prince Abdullah.
"The president is calling as part of his ongoing consultations with leaders around the world about the situation in Iraq, and he's also urging them to listen carefully to his speech at the United Nations," Fleischer said.
The United States argues Iraq's pursuit of weapons of mass destruction -- and suspected possession of some forms of such weapons -- poses a clear and imminent threat to regional and world peace.
National security adviser Condoleezza Rice, speaking on television Sunday, said the world could not afford to wait for definitive, incontrovertible proof of Iraq's possession of weapons, including nuclear arms, because the "smoking gun" could turn out to be "a mushroom cloud."
Washington's unilateral saber rattling has produced strong notes of dissent from allies in the Middle East and Europe, a situation Bush is now attempting to address in his domestic and international consultation prior to making a decision on action.
French President Jacques Chirac in a newspaper article Monday put forward a plan for a strict U.N. warning and 3-week deadline for Iraq to allow the resumption of weapons' inspections before any military action would be taken. The White House on Monday did not comment on the Chirac plan.
Bush was scheduled to meet Monday with Canadian Prime Minister Jean Chretien to discuss the implementation of a "smart border" program to tighten security along the frontier and yet not impede the heavy flow of trade across it.
It was widely expected Bush would also lobby Chretien on Iraq and the need for strong action, including military force if needed, but Fleischer Monday declined to indicate such.
Chretien has expressed caution on the question of military action. Canada's Deputy Prime Minister John Manley, speaking Sunday, said Canada would oppose any pre-emptive military strike by Washington.
"They'd be going in without Canadian support," he said.