In the letter, according to the New York Times, he appealed for more time to fend off charges of favouritism and cronyism.
Wolfowitz also asked to appear before the board next week in the interest of "fairness to me" and "good governance," the Times report said, relaying a description of the letter from people who had seen it.
It said many of the bank's 24 directors had been "taken aback by the tough tone of the letter," the contents of which have not been formally released by the World Bank.
A bank board committee is also investigating his hiring of former White House aides, also on generous pay deals, to work in his inner circle after he took over the World Bank in June 2005.
The New York Times said the board had been "puzzled and put off" by Wolfowitz's decision to retain a private lawyer, which has raised the stakes in the divisive row over whether he should resign as bank chief.
At a White House meeting on Wednesday about efforts to combat malaria, Wolfowitz won fresh support from the US president.
"I appreciate very much the fact that the World Bank is taking the lead in eradicating poverty in places like Africa, and Paul Wolfowitz, thank you for your leadership of the World Bank," George Bush said to his former colleague.
But European nations are continuing to step up pressure for Wolfowitz to quit.
On Wednesday members of the European Union's parliament urged the bloc's leaders to seek the resignation of Wolfowitz at an EU-US summit in Washington next week.
In a resolution, the lawmakers urged the EU's German presidency to make Wolfowitz understand that "his resignation would be an appropriate measure" given the bank's importance to fighting corruption.
The row is already undermining Wolfowitz's flagship campaign against corruption, according to a protest letter from staff involved in implementing the strategy on the ground.
The staff members wrote to Wolfowitz and the board appealing for "clear and decisive actions to resolve this crisis quickly," but did not call outright for the president's ouster.
"The credibility of our front-line staff is eroding in the face of legitimate questions from our clients about the bank's ability to 'practice what it preaches' on governance," they wrote.