Brooklyn Prospect Charter School students listen to a sports writer speak during February's Career Day.

The phone calls are bad, but the visitors are the toughest to reject.

That’s how Daniel Rubenstein feels about the admission requests that his charter school, Brooklyn Prospect, gets each summer from families who moved to the neighborhood after the school’s April lottery.

“This is a population that needs to be in a good school,” Rubenstein said. “Our school — which is a small, relationship-driven, intimate environment — would be better for someone that needs a community.”

But by law, Rubenstein must turn the families away. The state’s charter school law does not make provisions for schools to reserve seats for students who arrive to the city from far-flung locales after their April admissions lotteries. That means that charter schools, which are charged with serving the city’s neediest students, must exclude some of the students with the greatest need.

But after lobbying by Rubenstein and other charter operators, as well as by officials at the city Department of Education, one of the state’s charter authorizers is working on an option that would allow charter schools to open their doors in the middle of the year.

The mobile students — known as “over the counters” in Department of Education parlance — number more than 55,000 a year. They make up a student population larger than the total enrollment in Washington, D.C., San Francisco, or Boston.

Some of those students are transferring from one city school to another, according to city officials. Some are returning to the system after relocating temporarily, dropping out, or being incarcerated. And a large portion have never attended school in New York City before, city officials said.

The new students often have significant needs. “Many are students with disabilities, many are English language learners, and all of them are disadvantaged just by starting the school year later,” said Paymon Rouhanifard, the department’s executive director of school portfolio management.

About 30,000 of the new students are elementary- or middle-school aged, the grades served by most of the city’s 136 charter schools. But the responsibility of enrolling them has fallen entirely to district schools — exacerbating longstanding tensions over whether charter schools are fulfilling their mandate to serve the neediest students.

“One thing that is a breach between public schools and charter schools is that public schools have to take kids from their catchment area all year round,” said Morty Ballen, the CEO of the Explore Schools network of charter schools in Brooklyn.

Now, for the first time, a small group of charter school leaders and city officials are working to close that gap.

For about a year, Department of Education officials have been exploring ways to let charter schools enroll “over the counter” students. So have some charter operators, such as Rubenstein and Ballen. Last summer, Rubenstein sent a formal letter to the state requesting that officials examine how he could reserve a few seats for off-schedule applicants.

At first, the response that Rubenstein and city officials got was that only a revision to the state’s charter school law would allow them to take students over the counter. Under the current statute, “there’s really no way to hold seats that would allow kids to leapfrog” over families who applied through the regular lottery, Sally Bachofer, a State Education Department assistant commissioner, said earlier this month. SED authorizes some of the city’s charter schools.

Months passed without an update, Rubenstein said. Then, early last week, days after a reporter sent questions to his school’s authorizer, SUNY’s Charter Schools Institute, he got a call from the institute’s chief lawyer. The lawyer, Ralph Rossi, reported that he thought the law could be interpreted in a different way.

The new possibility would take advantage of the long waiting lists that 97 percent of city charter schools maintain.

All charter schools turn to the waiting lists between the lottery and the first day of school as some families opt for other schools. Some charter schools also use the lists to fill spots that open up when students leave over time, a process known as “backfilling.” But students who apply to a charter school after the lottery deadline are added to the end of the waiting lists, practically ensuring that they’ll never get in.

But what if the schools could shoot late arrivals to the top of the list? Already, charter schools can ask their authorizer for permission to give admissions preferences to groups of “at-risk” students, such as English language learners or students who require special education services. A school with those preferences in place would either reserve seats for those students in its lottery or give the students extra chances to have their names drawn.

SUNY lawyers think it could be possible to construct a new category just for students who come to the city after the regular admissions cycle, according to Cynthia Proctor, a SUNY CSI spokeswoman. When spaces open up in schools that have adopted the new category, those students would have the first crack at the seats.

Details about who the new at-risk category would include are “fluid at the moment,” according to Rouhanifard, who has been discussing the issue with the state for over a year. The category might privilege students who are new to New York State only, or just English language learners who are new to the city, or just families that arrive after Oct. 31, according to people involved in the discussions. All are agreed that any policy would have to safeguard against families who relocate temporarily for the sole purpose of gaming a charter school waiting list.

The fix would have to be refined and signed off on by SUNY and city lawyers before becoming an option for schools, which wouldn’t be until next year out of fairness to the families that applied in April under the current rules, according to Proctor. But if it passes legal muster, the change could go into effect for families that move to New York City from all corners of the globe in 2013.

It’s not clear what would happen if SUNY’s lawyers decide the change actually does not fit with state law. Rouhanifard said the city would consider pursuing legislation, which Rubenstein said he would be eager to support. But charter advocates say that with city charter schools’ future hinging on the next mayor, encouraging legislators to reassess a law that took a brawl to pass in 2010 isn’t high on the agenda.

“It’s important, but given all that’s going on it isn’t the most important thing,” said James Merriman, director of the New York City Charter School Center, about the issue of over-the-counter students. “It’s an interesting issue and one that my guess is we will see more discussion about over time.”

The discussion is likely to be hastened by growing scrutiny on charter schools’ mixed record of serving high-need students, such as students with special needs and English language learners. The state is in the process of setting enrollment “targets,” as required under state law, giving charter schools an added incentive to enroll more English language learners, whom they have so far failed to serve in large numbers.

While precise numbers are not available, officials say mid-year arrivals include a larger proportion of English language learners than in the city’s overall student population.

Even if the preference does become an option, there’s no guarantee that all charter schools would adopt it. The city would not require charter schools to admit mid-year students or fill spaces that open as students leave over time, as Denver and New Orleans have begun doing amid criticism that their charter schools were not serving the neediest students.

“That’s not a direction that we are moving in right now,” Rouhanifard said. “We respect the autonomy of charter schools.”

Without being required to accept mid-year arrivals, charter schools that do not “backfill” would continue not to accept students over the course of the year. (They might still set a preference that would allow them to take summer arrivals in a school’s entry grade.) A recent self-evaluation of the city’s charter school sector identified whether to “backfill” vacated spots as a major point of dissent among schools.

Rouhanifard said he had spoken to at least a dozen charter school operators interested in being able to accept off-schedule arrivals. But they represent a fraction of the city’s charter schools.

“Not every charter school wants to do it because it is a disruption,” Ballen said about figuring out how to accept the needy students who arrive mid-year. “I don’t know how many charters have the appetite to do it.”

But those that do won’t just be helping needy students, Ballen said. “If we can solve this, it [will] remove a wedge in the charter-public school community,” he said.

Leo Casey, a teachers union vice president, said the policy shift would have the most impact if prominent charter networks that do not currently “backfill” begin using over-the-counter students to fill seats that open up through attrition.

Still, Casey said, allowing charter schools to choose to accept the students “would be a step in the right direction.”