TRENTON — New Jersey Senate President Stephen Sweeney on Tuesday threatened to shut down state government next month, declaring he would block a budget that doesn't change the formula the Legislature uses to fund schools.

"The Senate is not going to pass a budget that doesn't start to address the unfairness in this funding formula," Sweeney (D., Gloucester) said at a rally Tuesday outside the Statehouse Annex, where he was joined by school officials and advocates from Cherry Hill, Robbinsville, Freehold, and other districts.

"We know this can't be fixed overnight," he said. "But it has to start now."

Sweeney's remarks may portend a showdown with Assembly Speaker Vincent Prieto (D., Hudson), with whom he has clashed on school funding.

The New Jersey constitution requires the state to pass a balanced budget by the end of the fiscal year, June 30. Both houses of the Legislature must pass the same spending bill to send to Gov. Christie. The governor can veto any line item.

The state passed a funding formula in 2008 but has not abided by it. The formula calculates a base cost of educating students and provides more money to districts with additional needs.

Sweeney has said he wants to increase funding for schools by $500 million over five years, eliminate so-called adjustment aid for districts that was intended to be temporary after the state passed the 2008 law, and reallocate money to districts where enrollment is growing.

Prieto on Tuesday called for a cooperative approach to address the issue.

"I've made clear that we need to tweak the school funding formula to fix adjustment aid and enrollment growth, while also improving special education and preschool access, and that I want to negotiate the new state budget without punishing school districts, playing politics or hurting low-income children," he said in a statement.

Christie also has pushed for a school-funding overhaul. Initially he called for equal per-pupil spending — which would boost aid for suburban districts at the expense of poorer ones — but then challenged Democrats to negotiate with him on a compromise.