The Loudoun County School Board might cut bus service for county students who attend Thomas Jefferson High School for Science and Technology in Fairfax County, a move that could jeopardize county students’ ability to go to the elite magnet school.

Loudoun schools officials are mulling over a number of potential cuts to address a projected $37.7 million shortfall in next year’s $950 million budget. The reductions under consideration include closing four small schools to save $2 million and cutting 33 positions for foreign language instruction to save $2.5 million.

A decision to cut bus service for the 200 students who cross county lines to attend TJ could have a significant impact on Loudoun’s participation in the school. The county’s relationship with Fairfax over the regional governor’s school has frayed in recent years. Loudoun has considered pulling out of the program entirely in an effort to keep the county’s brightest students at home and local tax dollars in Loudoun schools.

The School Board pays $390,000 to bus students to TJ, about 1 percent of the anticipated budget shortfall for next year. The county spends more than $3 million annually in tuition for its students to attend the magnet school.

“People always ask the School Board to make hard decisions during budget time with the ­caveat that they don’t make hard decisions affecting me,” said Wayde Byard, a spokesman for Loudoun schools. There are “nothing but hard decisions for the School Board right now,” he added.

School officials said 214 Loudoun students attend TJ this year, and they compose about 11 percent of the school’s 1,850 students. The majority of the Loudoun students board five county buses each morning bound for TJ. Without the funding for the bus service, the students would still be able to attend the school but would have to arrange their own transportation, Loudoun school officials said.

Vipin Jain, a Loudoun resident who has one son at TJ and another enrolling next year, said that cutting the bus service would discourage participation among county families.

“If you don’t offer the bus, there’s no way to get there,” Jain said. “So of course a lot of people aren’t going to go there.”

Byard said that parents whose children want to attend TJ are told by Loudoun officials that the school requires an added level of family commitment.

“They are told straight up that it’s a strain on time and resources,” Byard said. “It’s a great high school with a unique curriculum, but it comes at a price, and we stress that.”

Byard said that the School Board had not made final decisions regarding the budget and that it was unclear how soon the shortfall issue would be resolved.

Facing budget constraints in recent years, the county has debated whether continuing with TJ makes sense.

In 2013, Jack D. Dale, then the superintendent of Fairfax schools, drafted a plan calling on neighboring districts, including Loudoun County, to help pay for a $90 million overhaul of the TJ building near Alexandria. Loudoun School Board members openly questioned the county’s participation in the school. Later that year, then-Attorney General Ken Cuccinelli II (R) wrote an opinion stating that Loudoun did not have to pay for the renovation under Dale’s plan.

After Cuccinelli’s findings, Fairfax County revised the payment proposal’s legal language, and Loudoun later signed on.

Loudoun has been strengthening its advanced high school course offerings, and the School Board recently approved a proposal to construct a high school building that will house programs that focus on science, technology, engineering and math. But a significant percentage of the county’s top students leave for TJ. Of the 40 National Merit semifinalists from Loudoun this year, 15 attend TJ.

But the school system is years from forming a school that could compare to nationally renowned TJ, which offers courses in subjects such as oceanography, neuro­science and physics.