Mayor Bill de Blasio Thursday pledged to fight a budget proposal from Gov. Andrew M. Cuomo that he said would put the city on the hook for more than $1 billion in Medicaid and CUNY costs by 2020. But de Blasio dialed back his criticism after Cuomo said there are ways to ensure the plan doesn’t cost the city “a penny.”

Cuomo said efforts to reduce the costs of bureaucracy at the City University of New York and in the administration of Medicaid would offset any additional expenses for the city.

“At the end of the day, what you will see is it won’t cost New York City a penny, but we will make joint streamlining policy efficiency changes,” the governor told NY1.

De Blasio responded in a statement that City Hall will “commit to working with him to identify reforms and efficiencies.”

The mayor had earlier Thursday in Queens vowed to do “whatever it takes” to stop Cuomo’s plan to have the city cover the annual growth in its Medicaid expenses as well as one-third of the costs of CUNY colleges.

De Blasio, who has had a long-running feud with the governor, said the state budget pitch was “not fair” and “threatens our ability to serve our people.” The city currently pays virtually nothing for CUNY but appoints one-third of the college system’s board.

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Cuomo spokeswoman Dani Lever said for finding cost savings, the Medicaid and CUNY programs “are jointly managed so we need the cooperation of the city to make this possible.”

De Blasio had expressed some concern Wednesday after the governor unveiled his budget in Albany. After reviewing more details, he took a more forceful stance, only to soften his tone with a statement welcoming Cuomo’s update on how savings could be achieved.

Citizens Budget Commission vice president Maria Doulis estimated the costs to the city based on the budget Cuomo laid out Wednesday would be about $190 million for Medicaid in the next fiscal year, increasing to $478 million the year afterward.

Cuomo’s plan would offset the city’s responsibility for CUNY in the next fiscal year by paying for part of the professors union’s lapsed contract, but the city would face $485 million in fiscal year 2018, Doulis said.

Asked about the city and state working together to find savings, Doulis said Cuomo’s plan “suggests the budget figures are ‘targets’ for finding savings.”

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De Blasio is expected to deliver his formal response to Cuomo’s budget in two weeks.