ALBANY — Six bills with majority sponsors in the Senate and Assembly would allow school districts and local governments to raise property taxes above the current 2 percent annual cap.

“Despite the cap’s record, some lawmakers seem determined to weaken it,” said Tim Hoefer in a blog Wednesday for the independent, nonpartisan Empire Center, a fiscally conservative think tank. “In the first month of the 2017 legislative session, members filed 17 bills that would allow property taxes to rise faster by adding exclusions to the cap or changing the cap’s mechanics.”

Since 2011, school districts and local governments have had to stay within 2-percent annual growth in their budgets. If a school board or local government wanted to spend more, it would have to inform its constituency and then secure a supermajority of votes to exceed the cap. That provision is particularly difficult for school districts that put much of their budgets to a public vote, rather than simply a vote by the board.

Bills that have already gained majority sponsors in the Senate and Assembly would eliminate inflation as the ceiling for increased spending if inflation is less than 2 percent, in effect providing more spending room. Others would exclude costs such as school resource officers — usually a police officer — and other safety costs from spending under the cap.

Gov. Andrew M. Cuomo brought the 2-percent spending cap to New York. He and his supporters say it has kept taxpayers from paying an additional $7 billion in property based on the typical growth in local budgets before the cap was created in 2011.

Teachers unions and school boards influential with the Legislature have pushed for an easing of the constraint particularly in the face of rising pension costs and demands by state and federal policy makers to add instruction and other services.

Cuomo has strongly defended the cap and is proposing that local governments find ways to consolidate costs to save more.

Bills only sponsored by members of the Assembly’s Democratic majority would also make it easier for school boards to override the 2-percent cap without requiring the current super majority vote of residents.