As Rudolph W. Giuliani runs for president, his image as a chief executive who steered New York through the disaster of Sept. 11 has become a pillar of his campaign. But one former member of his inner circle keeps surfacing to revisit that history in ways that are unflattering to Mr. Giuliani: Jerome M. Hauer, New York City’s first emergency management director.

In recent days, Mr. Hauer has challenged Mr. Giuliani’s recollection that he had little role as mayor in placing the city’s emergency command center at the ill-fated World Trade Center.

Mr. Hauer has also disputed the claim by the Giuliani campaign that the mayor’s wife, Judith Giuliani, had coordinated a help center for families after the attack.

And he has contradicted Mr. Giuliani’s assertions that the city’s emergency response was well coordinated that day, a point he made most notably to the authors of “Grand Illusion,” a book that depicts Mr. Giuliani’s antiterrorism efforts as deeply flawed.