Tensions between unions for city teachers and police officers are heating up over a United Federation of Teachers directive telling school employees not to wear T-shirts to work backing the NYPD.

At issue is an online message circulated earlier this week to UFT members cautioning them against a grassroots members' plan by some to show the sartorial support for police on the first day of school -- and that violators could be reported to the schools chancellor.

The warning infuriated Patrick Lynch of the Patrolmen's Benevolent Association, already angry at UFT boss Michael Mulgrew for backing the Rev. Al Sharpton's anti-police-brutality rally held Aug. 23 in the aftermath of the chokehold death of Eric Garner in July.

"Mike Mulgrew needs to consider the opinions of the vast majority of his members before misusing their dues money to support anti-police issues," Lynch said in a written statement. "I also find it odd that a union would use management-like scare tactics in order to suppress the free expression of their members' ideas."

In his own statement, Mulgrew said that while he encourages his members "to express their opinions," Department of Education regulations "require school personnel to avoid distracting clothes and openly political statements when in school."

He added: "The Eric Garner march was a teachable moment for all New Yorkers -- but the lesson seems to have been lost on Mr. Lynch."

Asked Friday about the controversy, Mayor Bill de Blasio said employees have free-expression rights but unions can communicate with their members as they see fit. He disputed there was a battle brewing between cops and teachers, saying, "I'm just not interested in artificial conflicts. "