The Alexandria City School Board is considering a plan to send students to Fairfax County’s Thomas Jefferson High School for Science and Technology, the prestigious magnet program for Northern Virginia’s brightest teenagers.

Alexandria, with more than 13,000 students, is the only system among districts in the region without students attending TJ, as the elite school is known. About 20 percent of TJ students come from Loudoun, Arlington and Prince William counties and Falls Church and Fairfax City.

At two meetings last week, members of the Alexandria board’s advisory committee for talented and gifted programs urged school officials to approve a proposal to send students to TJ. They suggested sending 13 freshmen and one sophomore for fall 2014. By regulation, the number of students Alexandria could send to TJ is limited by the relative size of the district.

Adele Morris, chairwoman of the talented and gifted committee, told board members that Alexandria should take advantage of TJ, calling it “an incredible resource.”

Some board members bristled at the idea. “We have 13,000 kids that we have to worry about, and right now a lot of them are not doing that well and that has to be corrected,” said Patricia Ann Hennig (District C). “And to be honest with you, I don’t know that we’d have 14 kids pass the entrance exam.”

For the system, the decision could have budget implications for years to come. Fairfax County announced that its school system will seek payments from neighboring districts to help pay for a $90 million renovation and addition at TJ. Many school districts are considering whether to pull students from TJ rather than pay for the building outside their county’s boundaries.

Board member Kelly Booz (District B), who serves as the talented and gifted committee liaison, said she supported the proposal, noting that U.S. News & World Report ranked TJ among the top five high schools in the nation.

“When we have the number four public school in the country at our doorstep, I want to make sure that our students have the best opportunities that we can provide,” Booz said.

In a May letter to Alexandria City Superintendent Morton Sherman, Fairfax County Superintendent Jack Dale wrote that he would “invite and welcome Alexandria City Public Schools to participate in the regional school.”

Under the proposal, Alexandria schools would pay the Fairfax County system about $213,000 in 2014 to pay for the 14 students to attend TJ, which includes covering part of the renovation costs. By the 2017-18 school year, the Alexandria system would pay about $943,000 for possibly 53 students at TJ. The payment estimates do not include the cost of transporting students to TJ, which school officials said might require an additional $22,000 or $46,000 per year.

“We’re weighing our options,” said board member Christopher Lewis (District C), a 1997 graduate of TJ.

“There’s a lot of pride in T.C. Williams. A lot of folks are e-mailing us, asking, ‘Why would we pay to send students away to another division?’ ”