Richmond City Council appears to be stuck between a rock and a hard place as it seeks to craft a balanced $709 million operating budget that would become effective July 1.

On one side are passionate supporters of Richmond’s public schools who want the council to shift more local tax dollars into public education to avoid the potential shutdown of Armstrong High School and four elementary schools. Find the money, they say.

On the other side is a city administration led by Mayor Dwight C. Jones warning that cutting any more money could cause disruptions of city services and deeper financial problems.

Overhanging the budget talks — and adding to the council’s angst — are projections of a deficit of between $6 million and $9.6 million in the current budget and concerns that additional slashes by the council could ensure that expenses exceed revenues in the fiscal year that begins July 1.

“We are doing everything we can, but we have not been able to close the gap,” the city’s chief administrative officer, Selena Cuffee-Glenn, disclosed Monday. She reiterated that she plans to work with the council to consider options to end the year in balance, as state law requires.

On Thursday, April 28, the nine council members are to gather for another attempt to reach a consensus on changes to the mayor’s proposed 2017-2018 operating and capital budgets, mostly to increase school funding.

Providing even a portion of the $18 million that the Richmond School Board is demanding, though, is proving stressful — with a May 15 deadline looming to approve the education portion of the new city budget.

Councilman Parker C. Agelasto, 5th District, offered a reality check to school supporters during Monday night’s council meeting, telling them that the best the schools can expect is $4.97 million in additional funds, an amount that he and five other members support and that would be aimed at improving teacher pay.

Still, at this point, the council has been unable to reach agreement on possible cuts or revenue enhancements to provide that money. For example, only two members so far want to impose a 60-cent tax on cigarettes to raise a projected $5 million. Richmond is the only city in Virginia that does not tax smokes.

The stress that council members are feeling in finding extra dollars seemed to boil over during the public comment period. Council members started verbally pushing back when school supporters accused them of not caring. Several speakers insisted the city has plenty of money and only has to make adjustments to provide the public schools with $18 million more.

Councilwoman Reva M. Trammell, 8th District, fired back that it was insulting that anyone would believe that City Council is deliberately ignoring the educational needs of city children. She cited her proposal to provide new funding to reopen the now vacant Summer Hill Elementary School to provide relief from overcrowding in South Side schools.

Ms. Trammell claimed the School Board and its top officials are not providing the council with accurate information about the money it receives, creating mistrust and frustration.

“What happened to the $18 million that council provided last year?” she asked, unaware Mayor Dwight C. Jones’ administration had only recently released the money. “We’ve been told there are 9,300 vacant seats and now its 5,000. What’s with that? We need to know.”

When a school supporter expressed surprise that the council was not receiving the same updated and informative responses she received, Council President Michelle M. Mosby, 9th District, disagreed.

Not only is City Council not receiving timely information, Ms. Mosby said, “there are inconsistencies in the information” the council is receiving.

“We were told that Elkhardt Middle School was going to be demolished, but yet it is still standing,” she said. The lack of straightforward information is adding to council’s angst, she said, noting the school system has not reported how the extra $9 million the council provided during the current fiscal year has been spent.

Councilman Chris A. Hilbert, 3rd District, was equally trenchant about boosting funding.

“I want everybody to know that this body spent hours and hours and hours last year and found $9 million,” he said. He noted the school system would receive that same increase this year, but he joined Ms. Mosby in insisting that the council, too often, is left in the dark on where the money is spent.

He said that the council needs the cooperation of the School Board to ensure that “when the money is sent over for ‘x’ that it’s spent for ‘x.’ ”

No one from the school system’s administration or the School Board was in the City Council chamber when the remarks were made.

But afterward, School Board member and council candidate Kim Gray, 2nd District, said she had no idea what “the council members are talking about.”

“What inconsistencies?” she asked.

Ms. Gray said the School Board posts its spending online and has sought “to be as transparent as possible.”

Council had spent four hours earlier in the day on the budget, but left with nothing resolved.

Mr. Hilbert, who recently dropped out of the mayoral race and plans to run again for City Council, has tried to make it simple to find the money. But he has yet to win support for his plan to cut appropriations to departments by an additional 1 percent across the board to raise $4.7 million. That would be on top of the mayor’s 12 percent cut in spending for city agencies. The administration previously said departments cannot stand an additional reduction if council does not want services slashed.

Nor is there any agreement on a proposal to cut $1.7 million from the Richmond Ambulance Authority’s $4.5 million in savings to free up funds for schools.

Council’s staff on Monday offered 22 additional proposals for budget cuts to free up at least $2.6 million for members to reallocate. The proposals would roll back appropriations for a variety of items in which spending in recent years has fallen short of appropriations, ranging from overtime for the Fire Department and funding for plowing winter snow to payments for natural gas for the Richmond Justice Center and Public Works.

However, the administration appealed for time to address the fresh proposals, and is to respond at the session on Thursday.

The only good news came from the administration. Ms. Cuffee-Glenn acknowledged real estate taxes are projected to yield $1.7 million more than anticipated in the next budget. However, the mayor has yet to certify that increase, and until he does, the money cannot be recognized and used by the council.