Mayor Martin J. Walsh, who swept into office on the strength of labor, is facing a major test in the coming months as his administration launches into negotiations with nearly every employee union with a city contract.

All but two of Boston’s 40 collective bargaining agreements are expiring by fall, including many that ended in June.

Walsh campaigned for mayor with significant union backing, and many residents have raised concerns over labor’s role in his nascent administration. He had promisedto avoid the rancor and distrust that crippled talks between some city unions and the previous administration.

But as a new round of bargaining shifts into high gear, the mayor has to tread a fine line between negotiating affordable contracts with city police and firefighters while avoiding protracted and costly binding arbitration, political and fiscal watchers say. His labor negotiating teams must also satisfy the civilian unions — including the 8,200-member Boston Teachers Union — that are each angling for higher and equitable contracts.

“He owns this now,’’ said City Councilor Salvatore LaMattina.

Already tensions are simmering. Teachers held a “walk-in’’ last month to press for a fair contract “that is good for students, good for schools, and fair to our members,’’ said their president, Richard Stutman.

Stutman said union leaders, whose contract expires Aug. 31, have been meeting with the School Committee since February. He described the talks as slow, collegial, and to a certain extent productive, but “there are many serious outstanding issues on both sides,’’ he said.

Superintendent Tommy Chang and the School Committee are pushing for significant reforms to try to improve student achievement, including added measures that would attract and reward more quality teachers in the classroom.

The union is pressing for improvements, including better learning and teaching conditions and fair and equitable wages. Stutman also said the school district’s proposal on such things as not holding positions for teachers on six-month maternity leave is troubling.

“We find that backward, and very harmful to our members, our schools, and the students we teach,’’ he said in an interview.

David Sweeney, Boston’s chief financial officer, said the city’s labor officials will continue building on the work they began 2½ years ago, when Walsh became mayor. Since then, he said, the city and its unions have settled seven contracts — including with the powerful fire union —and negotiated an enhanced health insurance package that saves taxpayers money.

“We are looking to continue this trend,’’ he said.

He said city labor officials hope to preserve Walsh’s campaign pledge to work quickly and collaboratively with the city unions and avoid binding arbitration, used successfully by police and firefighters through the years to net hefty pay packages.

“Arbitration will never be this administration’s goal, and we are doing all that we can to negotiate agreements in the best interest of the city’s residents,’’ Sweeney said.

Walsh said he will be involved in some degree with the negotiations, but the bulk of the work will be led by his labor relations team. He said there have been offers “going back and forth with some unions,’’ but the process is still early.

“I feel confident that we are going to be able to get them at the table,’’ he said of the city unions. “I feel good about the process.’’

Samuel R. Tyler, president of the fiscal watchdog Boston Municipal Research Bureau, said the mayor must convey that he is both reasonable and practical, while also remaining firm on his resolve for reform and efficiencies.

Tyler said Walsh’s $2.8 billion budget does not have much financial wiggle room. Expenses for health insurance, debt services, and pensions are soaring.

What’s more, demands are high in the city’s four biggest departments: school, fire, police, and public works. Those accounts — plus state assessments for charter school tuition — represent 85 percent of the city’s total budget, Tyler said.

The question now facing the administration is how to keep the unions at the bargaining table. When talks broke down in the past, the police and firefighters sought resolution from an arbitrator at huge costs to the city.

Last year, an independent arbitrator ordered the city to give the Boston Police Detectives Benevolent Society a 28.4 percent pay raise over six years — beginning in 2011.

That decision — which the city estimated will cost $23 million — will further embolden other public safety unions to bargain for more, fiscal watchdogs said.

In 2013, a controversial arbitration award gave the Boston Police Patrolmen’s Association, the largest union in the police department, a 25.4 percent pay hike over a six-year period. That package is projected to cost taxpayers $87 million.

Both the detectives and patrolmen’s contracts expired June 30.

Those arbitration awards were preceded by another hefty arbitration award to firefighters in 2010. Tyler said the awards were “feeding off each other’’ and further escalating the cost of city contracts.

“Binding arbitration is broken, and there needs to be a new way to deal with impasse of public safety contracts,’’ he said. Leaders of both the patrolmen and detectives’ groups did not return calls for comment.

Walsh came into office when most of the contracts were settled under the previous administration. His labor team quickly resolved a long standoff and worked out an agreement with the Boston Firefighters Local 718 that included a lower pay-package than the police union’s.

The firefighters contract expires next year.

Walsh said he hopes to settle with the two big police unions quickly, but said it was difficult to predict whether a deal will be reached when those contracts expire.

“I’d love to see something happen,’’ he said. “But I’m not sure if it will happen exactly by the date. Our intension is to get there as quickly as possible and not drag on for years.’’

Walsh’s labor team must also wrestle with civilian unions, including members of the Service Employees International Union and Salaried Employees of North America.

Civilian unions, which do not have binding arbitration in their legal arsenal, must negotiate with the city. Previously, the talks have dragged on for years. When all attempts at a settlement fail, the city could impose its will and implement a contract as a last resort.

In the last go-round, the civilian unions, including the teachers, settled on a contract that gave them a 12.6 percent pay boost over a six-year period, beginning in 2010.

Elissa Cadillic, who heads the Boston Public Library’s largest union, said her union will press for bigger paychecks for its members, who include custodians, laborers, and library assistants in the American Federation of State County and Municipal Employees, Local 1526. Their average salary is $38,000, she said.

“As a low-wage organization at the Boston Public Library . . . none of us begrudge the arbitration award for the public safety unions. It’s fair,’’ she said. “Many of them do jobs that we don’t or can’t do. But we are all city employees. And we think that we deserve a fair and equitable wage as well.’’

Meghan E. Irons can be reached at meghan.irons@globe.com. Follow her on Twitter @meghanirons.