The Boston City Council moved Wednesday to examine the process for appointing constables after a 33-year-old man named to the position in April allegedly opened fire last week on two Boston police officers who were investigating a report of a domestic disturbance in East Boston.

A hearing to examine the vetting procedures was requested by District Councilor Timothy McCarthy, who said he was concerned that councilors weren’t informed of Kirk P. Figueroa’s criminal history outside of Massachusetts before they approved his application to become a constable.

“It concerns me that this perpetrator had a past, a past of arson, a past of impersonating a police officer,’’ McCarthy said during a council meeting. “Clearly the vetting process is broken and we need to fix it before anything else comes before our desk.’’

The request received unanimous support from the council and was referred to the Committee on Public Safety and Criminal Justice. A hearing date has not been set.

Authorities say Figueroa was heavily armed Oct. 12 when he began shooting at officers who were investigating a report of a fight between him and his roommate on Gladstone Street at 10:41 p.m. Wounded in the gunfight were Boston police Officers Richard Cintolo and Matthew Morris.

Cintolo was released Wednesday from Massachusetts General Hospital, Boston police announced on Twitter. Morris is recovering at the hospital.

Figueroa was killed during the confrontation.

In the wake of the encounter, details emerged about Figueroa’s legal troubles in Georgia, where authorities say he set his vehicle on fire in 2009. He was also accused of unlawfully representing himself as a private detective while not holding a license, according to court records. Figueroa did not have a criminal record in Massachusetts.

The council approved his constable application April 13 on a voice vote and without any debate. The city said it has 142 sworn constables.

Constables are authorized by law to serve legal documents in civil matters like lawsuits, divorces, and evictions, and in some instances they can take people into custody and secure property.

Applications for the position are reviewed by Boston police, who conduct background checks on candidates, the city said. Police then send a list of qualified candidates to the City Council, which must confirm the applicants before they can be appointed to a three-year term by the mayor, according to officials and the city ordinance governing the position.

Applicants must live in Boston.

Speaking last week, Boston Police Commissioner William B. Evans likened constables to a “mailman’’ and said his department conducts a “general’’ background check into candidates. The inquiry is not as intense as background checks for people seeking a license to carry a firearm, he said.

Figueroa wasn’t licensed to carry a firearm, according to Evans. Under department rules, constables are not automatically granted firearms licenses or given preference if they apply.

Boston police said Wednesday no one was immediately available to discuss how constable candidates are vetted.

Laura Crimaldi can be reached at laura.crimaldi@globe.com.