ALBANY — In an emerging sign of possible budget-battle lines, Democrats in the New York Assembly rejected proposals on Monday by Gov. Andrew M. Cuomo linking an increase in statewide school aid to an array of his proposed changes. They also sided with Mayor Bill de Blasio’s bid for longer-term mayoral control of New York City’s schools.

The Democrats’ stance, articulated in several proposals released by Speaker Carl E. Heastie of the Bronx, keeps a tight focus in Albany on the issue of education, which Mr. Cuomo has promised to make a focus of his nascent second term.

In January, Mr. Cuomo, a Democrat, introduced a series of education proposals, including making a teacher evaluation system rely more heavily on state tests, raising the number of charter schools in the state and allowing failing schools to be taken over by outside groups. That position was criticized by teachers’ unions but praised by supporters of the charter school movement.

Mr. Cuomo’s education plan also includes a financial carrot: an increase of $1.1 billion in state aid if his proposals become law. But on Monday, Mr. Heastie suggested that such an approach would not pass muster in the Assembly, which Democrats control.