Mayor Bill de Blasio and Corey Johnson, the City Council speaker, shook hands Monday on a $89.2 billion city budget that included a major concession by the mayor to provide funding for discounted subway and bus fares for some of the poorest New Yorkers.

The budget continues the sustained rise in spending that has come on Mr. de Blasio’s watch. It is $4 billion more than the previous budget, including $254 million more for the Metropolitan Transportation Authority, called for by Gov. Andrew M. Cuomo and initially resisted by the mayor.

The budget was the first that Mr. de Blasio had to negotiate with Mr. Johnson, who was selected as the Council speaker in January on a pledge to make the city’s legislative body more independent of the mayor. The Council is expected to approve the budget on Thursday.

Mr. Johnson had pushed hard to get the mayor to agree to include funding for subsidized MetroCards, an initiative known as “Fair Fares.” The program will begin in January, with New Yorkers whose income is below the federal poverty line — about $25,000 a year for a family of four — qualifying for MetroCards at half the regular cost. It is estimated that about 800,000 people could be eligible for the subsidy, although far fewer are expected to enroll in the first year.