Mayor Bill de Blasio says that a “proactive” approach to governing will help solve the city’s structural budget deficit.

“We think a lot of the proactive strategies to save money in the long run,” he said in response to a reporter’s question about his plans to tackle future budget gaps. “Keeping people out of our jails saves money. We can help to make people who might not have been law-abiding citizens into law-abiding citizens. That also saves money. We can keep folks who would have been homeless from going into our shelter system. That saves money.”

Sources said members of Mayor de Blasio’s transition team floated possible posts to Liu, including Taxi & Limousine commissioner and finance commissioner.

But political observers say Liu still has his eye on elected office and has been exploring another run, possibly for Congress.

In his first two months in office, Mayor Bill de Blasio has found himself repeatedly at odds with New York’s governor, Andrew M. Cuomo. And at every turn, Governor Cuomo has not only stymied the mayor, but also seized the moment for his own gain.

The senior Cuomo administration official said Mr. de Blasio was continuing to seek the tax increase because he wanted to be able to tell his political base that he fought to the bitter end for a worthy cause, regardless of the outcome.

Anticipating that a new administration would drop Mr. Bloomberg’s suit, unions representing police officers filed their own challenge to the law along similar grounds.

Whether or not to continue that suit could figure into current contract negotiations between the police unions and the city, union officials said.

Schools Chancellor Carmen Fariña says charter-school students she booted from city-owned space in Harlem “are on their own” and can always go back to regular public schools.