The San Francisco Police Officers Association has begun canvassing for a controversial measure on the June ballot that would loosen the department’s policy on stun guns when officers start getting the weapons this year.

But anyone handed the latest Proposition H campaign literature wouldn’t necessarily know that.

Supporters are painting the measure as an effort to arm officers with Tasers, despite the fact that there’s already a plan in place to do that. At the same time, the police union’s leadership is telling the rank and file that Prop. H is about loosening the rules for when officers can use the weapons.

That strategy isn’t sitting well with the opponents, who are being significantly outspent in the campaign and say the strategy is deceptive.

“The Police Officers Association is lying to voters about what this is about,” said John Crew, a retired lawyer for the American Civil Liberties Union and leader of the No on Prop. H Committee. “They are saying one thing to their leadership, and other thing to the public.”

The June 5 vote on Prop. H is only the latest in a years-long fight over arming San Francisco police officers with stun guns. Since they were approved by the city’s Police Commission last year, the fight has shifted to how Tasers will be used.

Mayor Mark Farrell, a longtime police union ally, and Police Chief Bill Scott both oppose Prop. H, and three city leaders who endorsed it are offering only lukewarm support.

Weeks before the measure was filed by the police union in November, the city’s Police Commission voted to arm officers with stun guns. In March, the commission voted on a policy on their use with guidelines stricter than those spelled out by Prop. H, clearing the way for officers to be armed with the weapons this year.

But any hopes by opponents that the police union would ease up on its effort to pass Prop. H ended when supporters last weekend began knocking on doors and passing out flyers in the West Portal neighborhood.

“This June, vote to equip San Francisco police officers with Conducted Energy Devices commonly known as Tasers,” the campaign literature reads. There is no mention of the Police Commission vote.

But what’s really at stake, Police Officers Association President Martin Halloran wrote to his officers in the monthly POA Journal, is the policy language on when officers may shock suspects.

Halloran wants officers to be able to use Tasers when a suspect is “actively resisting” versus the current policy to use Tasers when a person is “violently resisting.” That policy was approved by the commission and vetted through several community working groups.

Opponents say that standard isn’t strong enough, and an action like holding onto a light pole could be interpreted as “actively resisting.”

Halloran, nevertheless, is pushing hard for the new standard.

“We believe the policy we put forward is reasonable and consistent with our own Sheriff’s Department and with the Oakland and San Jose Police Departments,” he said in an interview. “We believe this is going to reduce injuries for officers.”

Perhaps the most controversial element of the measure is that the policy could be amended only at the ballot box or with a four-fifths vote by the Board of Supervisors.

“They’re choosing to spend an enormous amount of money to take away the power of the chief and the Police Commission to regulate how a dangerous weapon is used,” Crew said. “This is completely unprecedented for any police union in the country.”

And when it comes to campaign cash, supporters of Prop. H are flush, according to the latest public filings with the San Francisco Ethics Commission. With most of the money coming from the police union, supporters have raised $221,013. That’s compared with the just under $41,000 raised by the No on Prop H group.

The police union may need every dollar it has to persuade city residents to vote for the measure that’s been widely criticized by city leaders.

Farrell supported the measure at first, but pulled his endorsement for Prop. H, calling it “unnecessary” after the Police Commission approved a policy.

Some groups and three city politicians still support the measure, according to the police union.

State Sen. Scott Wiener endorsed Prop. H before the Police Commission vote and said he is sticking with it.

“Once I give my word and make an endorsement, I typically don’t pull back,” he said, without mentioning the merits of the proposed policy under Prop H.

District Eight Supervisor Jeff Sheehy said he’s sticking with his endorsement too, writing in an email, “I am still supporting Prop. H.” Sheehy did not respond to a follow-up email asking why he supports it and didn’t return phone calls.

District Two Supervisor Catherine Stefani, who’s listed as a supporter on the police union’s Prop H. literature, did not respond to phone calls or emails from The Chronicle.

Evan Sernoffsky is a San Francisco Chronicle staff writer. Email: esernoffsky@sfchronicle.com Twitter: @EvanSernoffsky