The Fairfax County Board of Supervisors voted Tuesday to advertise a cap on the real estate tax rate that, if fully reached, would add an average of $100 to annual tax bills — on top of the extra $332 on average that homeowners in the county will see because of higher real estate assessments.

The 8 to 2 vote, which begins a weeks-long public process before a new tax rate is formally adopted in April, is an attempt to find some flexibility in bridging a gap in funding for the county’s school system.

School officials are seeking $63.8 million more in funding than what County Executive Edward L. Long proposed last month in a budget that is based on the county’s current tax rate of $1.085 per $100 of assessed value.

A 2-cent cap increase to $1.105 per $100 of assessed value would give Fairfax officials some room to raise real estate taxes in the county, which has 1.1 million residents, potentially providing $43.8 million in extra revenue if the new rate were to go that high.

However, board members made clear that they are not likely to approve such an increase and vowed to seek other ways to help fund the steadily growing county school system of 184,000 students.

“It is this chairman’s intention to seek ways to adopt a budget with a tax rate that is less than a 2-cent increase,” said Chairman Sharon Bulova (D), reading a prepared statement before proposing the new tax rate cap.

With Karen Garza, the superintendent of Fairfax County schools, and several School Board members and parents in the audience, the other supervisors took turns agonizing over the funding predicament the county finds itself in after school officials adopted a $2.5 billion budget that cuts 731 positions while seeking to give teachers competitive salaries with $41 million in raises.

Several supervisors said predictions by school officials of dire cuts and larger classrooms if the county decides against granting a funding increase of $98.1 million place an unfair burden on county homeowners, who are facing more taxes and fees elsewhere.

“People are reaching their limits here,” said Supervisor Linda Q. Smyth (D-Providence), who voted against the cap increase along with Supervisor Pat Herrity (R-Springfield).

“With the additional assessments, on top of the possibility of a tax increase, some of them are going to be pushed to their limits and beyond,” Smyth said.

Wrestling with that prospect, some supervisors suggested introducing a meals tax on restaurants in the county, a fee that exists elsewhere in Northern Virginia but that has never won support among Fairfax voters.

Others blamed school officials for not budgeting prudently enough, while still others accused state officials of not providing enough education dollars to Fairfax County.

County officials are hoping for $27 million to $32 million in state funds geared toward competitive salaries for teachers — which, along with a projected budget surplus of $4.64 million, would close some of the funding gap for Fairfax schools.

“We wouldn’t even be in this situation if we had gotten a fraction more from the state,” said Supervisor Jeff C. McKay (D-Lee), complaining about how state tax dollars are allocated in Virginia. “Yet the community has missed the entire point because all the guns are pointed at us.”

School officials said they were pleased that the county supervisors allowed more funding flexibility and added that they are eager to find some solutions before the county votes on a transfer of school funds by May.

“It allows the conversations to continue,” Garza said. “We’re hoping for the best in the end. But there are some difficult challenges ahead.”