Bennett said on Wednesday: "Mr Wolfowitz will not resign under this cloud and he will rather put this matter to a full [board] vote than to capitulate on his integrity."
European members led by France, Germany and the Netherlands have said Wolfowitz should step down to salvage the bank's credibility, which they say has been damaged by his handling of a high-paying promotion for his girlfriend, Shaha Riza.
"Wolfowitz must resign because... his act of favouritism for personal interest was wrong"
Ibby, Mumbai, India
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A board panel found his efforts on her behalf broke bank rules and represented a conflict of interest, but Wolfowitz says he acted on the advice of a board ethics committee and wants it to acknowledge its own failures.
In a last ditch plea to save his job, Wolfowitz appeared before the board on Tuesday, saying: "You still have the opportunity to avoid long-term damage by resolving this matter in a fair and equitable way that recognises that we all tried to do the right thing, however imperfectly we went about it."
A US request for an adjournment of the board meeting on Wednesday sparked speculation within the bank that its backing for Wolfowitz was softening.
While Tony Snow, the White House spokesman, said "we stand by our support of Paul as the World Bank president", one source said discussions moved to how Wolfowitz might resign once the meeting resumed.
However, the board later adjourned until Thursday without a decision.
Under a contract he signed in June 2005 when he became World Bank president, Wolfowitz would receive a year's salary, or around $375,000, if his services were terminated by the board or if he resigned.
Pressure to resign intensified on Wednesday as European countries signalled they would resist a bid by the Bush administration to keep Wolfowitz in the job.
The US had tried to a cut a deal that would have separated consideration of his ethics violations from a decision over whether he had the credibility to continue, but only Japan out of the G7 countries sided with Washington.