Tony Fratto, White House spokesman, said: "The president is going to try to select the individual he thinks is the best person for the job."
The World Bank board began meeting on Friday to discuss the issue.
"He should donate his year's salary, that he gets as a parting gift, to a Baghdad hospital"
Nick, Hampstead, USA
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The agenda was set to include how a new president should be chosen and whether an interim leader should be appointed to take over after Wolfowitz departs and before a new head is named.
The US, the bank's largest shareholder, has named the World Bank chief since the bank's inception more than 60 years ago.
Many critics have said that practice should be revamped in the wake of Wolfowitz's rocky two-year tenure at the bank.
Henry Paulson, the US treasury secretary, said on
hursday that he would help Bush identify a nominee after consultations with other World Bank member countries.
"I will consult my colleagues around the world as we search for a leader who will continue to focus the bank on creating opportunities for the world's poorest," Paulson said.
He made clear it would be an American. He said: "I see no reason why this should change and I see every reason why it's important that the World Bank should continue to be run by an American."
European countries, who by tradition always choose the head of the bank's sister organisation, the International Monetary Fund, have not challenged that long-standing practice.
Bank of Blair
While the White House cautioned that it was too early to speculate over a successor, names being tossed around in the US media included Paul Volcker, former federal reserve chairman; Robert Zoellick, former deputy US secretary of state; and Robert Kimmitt, deputy US treasury secretary.
Rumours have circulated in past days that Tony Blair, the outgoing British prime minister, could be tapped for the post.
Joseph Stiglitz, a Nobel prize-winning economist and former senior vice president and chief economist at the bank, said that Blair "is one of the people that is clearly being discussed."
Wolfowitz's decision to step down at the end of June has not quelled staff anger.
The bank's employee association has demanded the former US deputy defence chief and the Iraq war's architect be put on administrative leave immediately and be forbidden from making any policy decisions.
Senior bank managers urged staff on Friday to focus on the bank's mission of fighting poverty.