The Senate this week will debate legislation to lift the cap on charter school enrollment in some school districts, entertaining a bill that is significant different from a bill that has passed the House and which includes a measure to tie a cap lift to state reimbursement funding.

With just three weeks remaining in the session for lawmakers to pass controversial legislation, the chasm between the House and Senate over how to deal with charters could challenge their ability to get a bill to Gov. Deval Patrick’s desk that he would sign.

Sen. Sonia Chang-Diaz’s bill, which was released Friday from the Senate Ways and Means Committee and marked for debate next Wednesday, proposes to incrementally lift the cap on charter school spending through fiscal 2022 in the lowest performing school districts. The annual cap lift would be tied to full funding for the state’s charter reimbursement program to local school districts.

Chang-Diaz, a Jamaica Plain Democrat and the co-chair of the Education Committee, floated the idea back in May as a potential compromise to ensure that local school districts aren’t financially hurt by growing charter school enrollment. House Education Committee Chair Rep. Alice Peisch and charter school advocates deemed it a non-starter in negotiations, and the House eventually passed a bill that did not speak to charter reimbursement funding.

“This bill carves out a third way: where we serve the interests of students in both the charter system and the district system - rather than pitting them, and their families, against one another,” Chang-Diaz said in a statement to the News Service. “It addresses inequities in funding and student population, and allows strong-performing charters to expand.”

Chang-Diaz was not available to discuss the legislation on Friday, but said in her statement she was “doing everything I can to answer lingering reservations colleagues have about charters’ impacts on districts.”

Senators have until Monday at noon to submit proposed amendments.

“We’ll see,” said Senate Majority Leader Stanley Rosenberg, after presiding over the Senate’s session Friday. “We have a lot of work to do in the next three weeks and the Senate president committed it would be on the floor and it’s scheduled to be on the floor and will be fully debated and vetted by the members.”

According to the Massachusetts Municipal Association, the charter school reimbursement program was fully funded in fiscal 2014, but the fiscal 2015 budget provides $80 million, which will create a $33 million shortfall.

The Race to the Top Coalition, which has been actively pushing for charter school expansion this session, previously opposed efforts to link a cap lift to annual state funding even though the organizations support fully reimbursing school district for student costs, according to the formula.

Keith Mahoney, spokesman for the coalition, declined to talk about specifics in the bill, but said the coalition was pleased to see a bill scheduled for debate and said he looked forward to having “constructive” conversations with senators over the coming days.

The bill (S 2262) also proposes to create a new classification of schools known as “challenge schools” that fall among the lowest 20 percent for student performance and are on the cusp of being labeled underperforming. The superintendents would be required to convene local stakeholders and develop turnaround plans for their schools that would be submitted to the local school committee and teachers for approval.

Under the legislation, school districts and charter schools would be prohibited from expelling a student for poor academic performance, and charter school operators looking to open another school or expand would be denied if their student attrition rates exceed the three-year averages of the school district supplying the charter’s student population.

While the proposed cap lift in school spending on charter school enrollment mirrors the House proposal to increase by 1 percent a year starting in fiscal 2018 to 23 percent, the annual cap lift would be suspended if the Legislature fails to appropriate enough money for school district reimbursements.

Lawmakers have recently waited until the end of the year to determine if the state has sufficient surplus revenue to fully fund the reimbursement formula, and this year was no different with Gov. Deval Patrick signing a $36.5 billion fiscal 2015 budget on Friday that does not fully fund charter reimbursements.

The Ways and Means Committee estimated the legislation, when fully implemented, would cost between $5 million and $8 million and would be borne by existing appropriations in the fiscal 2015 budget.