Two candidates to replace Mayor Marty Walsh have been critical of his handling of suspended Boston Police Commissioner Dennis White's appointment. But others have carefully avoided criticizing the man they hope to replace — even as they have vowed accountability, transparency and reform in city policing.

The Boston Globe first reported on the allegations after White's appointment, which came in the wake of former commissioner William Gross unexpectedly resigning in late January. Walsh quickly suspended White and named an attorney to investigate the allegations. That investigation has proceeded out of public view, and without a precise end date, despite the mayor's past champining of police transparency efforts.

GBH News canvassed the candidates who hope to replace Walsh once he is confirmed as President Joe Biden's secretary of labor, likely next week.

In the past, City Councilor Andrea Campbell, who represents parts of Dorchester and most of Mattapan, has offered the most pointed criticism of Walsh, calling the vetting process for White “a failure.” Campbell’s campaign said if elected, the councilor would ensure a thorough and transparent process for selecting a police commissioner, but did not spell out further details.

At-Large City Councilor Michelle Wu, who has also been directly critical of Walsh, said the commissioner’s job is one of Boston's most important and that it required public confidence in the appointee's ability to promote internal change.

“It’s central that the police commissioner and the process of appointing that police commissioner builds trust with our communities,” she said. “What was missing in this case was a whole set of internal and external checkpoints and verifications ... Boston should have a national search that centers the voices and the needs of our local residents and community members who have been highlighting concerns and issues for decades.”

Walsh, through a spokesman, last month conceded that the process that led to White’s appointment should have been more “thorough.”

That's an admission that could be turned to political advantage. But, so far, most candidates for Walsh's job have been circumspect.

Council President Kim Janey has yet to declare in the mayoral race, but she will serve out the short balance of Walsh’s term when he departs for Washington, D.C., either way. After the allegations against White became news, Janey said she favored the independent investigation Walsh initiated. She didn't go much beyond that when she told GBH News that her focus “is on properly serving the people of Boston — and that includes ensuring accountability and transparency in city operations."

At-Large Councilor Annissa Essaibi George, like Walsh, is a Dorchester native. She's been clear that she would not be running for Walsh's job if he were not going to Washington. She said the independent investigation is the proper course of action. Boston, she said, “has the opportunity to lead on reforms and demonstrate the benefit of community policing, transparency and accountability.”

“As mayor, I will push for critical reforms,” Essaibi George said, including “expanding implicit bias training, getting mental health clinicians in every precinct, reforming the gang database, stepping up community outreach, supporting cadet program efforts and ensuring the department is more diverse and reflective of all the communities it serves.”

Like Janey, Essaibi George avoided critcizing Walsh, either explicitly or implicitly.

So did John Barros, Walsh’s former economic development chief. Barros promised to develop a “comprehensive, thoughtful and transparent approach” to choosing a new police commissioner. “That way, the people of Boston can be assured that anyone who takes that oath of office moving forward has been thoroughly vetted with the community’s priorities taken into account."

The candidate who walked the finest line was state Rep. Jon Santiago, D-Suffolk, an ER doctor by training. Santiago expressed support for Walsh’s investigation but put some distance between himself and the vetting process while also maintaining a tone of reform-minded collegiality toward the Boston Police Department.

“I support letting the investigation continue," Santiago said, "but at some point, we need a clear path on who will lead the BPD with transparency and accountability,”

“In the ER," Santiago continued, "we assess complex situations under intense pressure, make diagnoses and execute a plan … As mayor, I will establish a rigorous vetting process that brings qualified candidates to my desk only after seeking broad-based community input to ensure candidates have the trust and support of the communities they will serve. As someone who has given his life to public service, I have great respect for our law enforcement officials and look forward to working with them as mayor and improving the way we pursue justice in our city.”

When it comes to picking a city's top cop, Chuck Wexler, executive director of the Police Executive Research Forum, said Boston, like New York City, tends to follow its own rules.

According to Wexler, who has assisted with dozens of police leadership searches, the two municipalities are unique in their lack of formal search processes and limited selection pools.

Margo Frasier, vice president of the National Association for Civilian Oversight of Law Enforcement, said that while her organization does not have specific guidelines for choosing police leadership, community input, however it manifests itself, should be prioritized.

“[Police leaders] are there to serve the whole community,” she said in a phone interview. “Community should be involved in the selection and recommendation process.”

For now, Acting Commissioner Gregory Long sits in the Boston commissioners' chair.

This article has been updated to clarify the Campbell campaign's comments on selecting a police commissioner.