Karen Lewis, president of the Chicago Teachers Union, addresses a crowd during the 2012 strike. (Charles Rex Arbogast/AP)

Chicago Teachers Union members plan to begin voting Wednesday on whether to authorize a strike, setting in motion the latest in a series of political challenges for Mayor Rahm Emanuel.

The teachers’ contract expired in June, and efforts to reach a one-year contract agreement with Chicago Public Schools fell apart during the summer amid differences over teacher evaluations, among other issues. Talks between the union and the nation’s third-largest school system restarted in August, but the two sides have been unable to reach a compromise on a number of issues, including staffing, salaries, pension contributions and health-care benefits.

“Negotiations are stuck in neutral, and it’s not clear that anyone on the other side of the table sees any problem with that,” said Jesse Sharkey, the union’s vice president, who added that the sides are no closer together now than they were three months ago. “I haven’t seen evidence of good faith in really trying to grapple with the problem of under-resourcing public schools, the problem of very deep cuts and systematic deficits,” Sharkey added. “We’re saying it’s time to move this process.”

According to the union, the Chicago school board — whose members are appointed by the mayor — has proposed changes that would amount to a 12 percent cut in total compensation over a three-year term.

The school board also has rejected union proposals for smaller class sizes and more librarians, counselors, social workers and art and music programs, according to the union, as well as proposals that would cost little to nothing, such as doing away with redundant testing and giving teachers more autonomy in grading student work.

Chicago Public Schools officials said it is premature for the union to threaten a strike because the mediation process is not complete and the mediator involved in the talks has not yet heard discussions on a number of important issues.

The school system officials said the union’s proposal would cost an additional $1.5 billion in spending, which they call unrealistic for a district in a serious financial crunch. Last summer, the district cut about 1,400 jobs — including 500 teaching positions — and school system leaders have said that thousands more layoffs might be necessary.

“CPS is looking for solutions to our $1.1 billion budget crisis, and a strike isn’t the answer,” said Emily Bittner, a school system spokeswoman. “We’re negotiating in good faith with CTU leadership to reach a fair, multiyear agreement that protects teachers, their jobs and our classrooms — and we encourage CTU to join with us in Springfield to fight for equal funding for Chicago children.”

Union leaders say the vote will last at least three days or perhaps longer, if that’s what it takes to make sure that all 27,000 members vote. Under state law, 75 percent of the union’s active members must vote yes in order to authorize a strike.

It would be the first step in a months-long process that would have to play out before teachers actually decide to walk off the job.

Three years ago, Chicago teachers went on a seven-day strike that helped them win significant concessions, including a reduction of the proportion of their evaluations that would be based on standardized test scores. The strike also drew national attention to the bruising battle between the teachers union and Emanuel, and this week’s strike vote comes as Emanuel hardly needs another distraction.

This year, he weathered the indictment of his handpicked schools superintendent, Barbara Byrd-Bennett, who in October pleaded guilty to taking kickbacks from a $20 million no-bid contract.

Now, since a video was released of an officer fatally shooting 17-year-old Laquan McDonald, Emanuel is facing questions about a lack of accountability among Chicago police.

McDonald was running away from officers when he was shot 16 times, according to dashboard-camera footage city officials released only after a judge ordered them to do so. Emanuel fired Chief Garry McCarthy, but now the Justice Department has launched a broad civil rights investigation into the Chicago police.

Emanuel’s press office did not respond to a request for comment on the teachers’ strike vote.