Paul Wolfowitz told reporters on Monday that "we're very close to having that package put together," referring to an agreement that would decide how to fund a plan to cancel the debts.
Wolfowitz was speaking at a joint news conference with Gerrit Zalm, the Dutch finance minister, in The Hague.
He said that progress had been made since World Bank governors met in September.
The Group of Eight - the world’s seven leading industrialised nations plus Russia - had agreed at a summit last year in Scotland to write off $38 billion of debts owed by the world's poorest countries to the World Bank and International Monetary Fund (IMF).
The Netherlands is the fifth largest contributor to the World Bank and has fought hard to make sure that debt relief does not compromise the bank's ability to keep lending.
The G8 has not yet found a way to recapitalise the World Bank's International Development Association (IDA) to help cover the deficit from the debt write-off, which is due to take effect on 1 July. The IDA is the world's largest lending facility for poor countries.
"What we're looking for is the really solid commitments that will make this debt relief package ... really effective and make sure that the resources that are freed up are replaced by the commitments from major donors to provide additional assistance"
Paul Wolfowitz, the World Bank president
"Our goal is to have this done so that the debt relief will go into effect on 1 July as planned," Wolfowitz said.
"What we're looking for is the really solid commitments that will make this debt relief package ... really effective and make sure that the resources that are freed up are replaced by the commitments from major donors to provide additional assistance."
At a meeting of G8 finance ministers in Moscow earlier this month, Russia failed to convince its partners to allow it to buy back billions of dollars of Soviet-era debts in return for its creditors using that money to recapitalise the IDA.
US officials have said they want to see the World Bank move faster to match the action of its sister global lending organisation, the IMF, which announced in December that it was granting 100% debt relief to 19 countries that qualified for it.
However, Wolfowitz said the world's largest development lender and the IMF needed more reform.
He said he wanted to see if there was "harmful duplication" or gaps between the jobs of the two organisations.
"A lot of people are raising questions about the changing role of both institutions," he said.
"I don't exclude changes in the governance system as well, but I would say let's start with figuring out what our mission should be today, where we want it to be 30 years from now."