Aides to Mr. Bloomberg, a political independent, have not attacked Mr. de Blasio, but say it is their responsibility to correct the record when the mayor or his supporters misrepresent it.

“We’ve been surprised the extent to which it’s been necessary,” Mr. Wolfson said.

Administration officials have come to expect resistance from foes old and new on certain policies. As Capital New York reported, some education reform advocates have for weeks peppered reporters with opposition research on Mr. de Blasio’s prekindergarten program.

But the more frequent public defenses of Mr. Bloomberg from his own staff represent a new front.

In a statement, Mr. Ragone said, “Anyone who has worked here at City Hall should know that if you want to run an effective government, you just call ’em like you see ’em and move on to fixing the problems.”

“You just don’t have time for petty political skirmishes,” he added.

Many jabs at Mr. Bloomberg this year have touched on prominent campaign themes, like income inequality and the stop-and-frisk policing tactics — an issue that Police Commissioner William J. Bratton said in January had “more or less” been solved, noting that stops had fallen precipitously during Mr. Bloomberg’s final years in office. (The figure has continued to drop in areas across the city this year.)

The administration has lamented the unresolved labor contracts under Mr. Bloomberg. It has referred less often to the surplus that left Mr. de Blasio “in very good shape for his first budget,” said Carol Kellermann, the president of the Citizens Budget Commission.

Earlier this month, Mr. de Blasio criticized the Bloomberg administration’s Hurricane Sandy relief plans as lacking “a focus on real human beings.” Days later, he issued an announcement celebrating the city’s new “.nyc” websites — championed by Mr. Bloomberg and Christine C. Quinn, the former City Council speaker who ran against Mr. de Blasio in the Democratic primary for mayor last year — without mentioning either of them.

On housing policy, where Mr. de Blasio has spoken of a significant break from the Bloomberg years, the new administration has also strained to identify differences on some projects. At a groundbreaking in April for the first phase of an affordable housing development in East New York, Brooklyn, Mr. de Blasio said he planned to count 278 units from the complex, negotiated under Mr. Bloomberg, toward his goal of 200,000 affordable units in 10 years.