Anderson said he would also like the commission to explore the issue of female officers who contend that are harassed and intimidated by male supervisors.
At least four police officers have filed suit against Washington, 44, accusing him of sexual harassment, retaliation and discrimination.
The city settled two of the lawsuits last year for a total of $75,000.
Washington, a 24-year veteran, has been the subject of 14 citizen complaints, most of which have accused him of physical abuse. Internal Affairs investigators deemed the complaints unfounded.
Settlement talks, however, are scheduled for December for a lawsuit filed by a South Philly man who alleges that Washington split his head open with a baton in 2010.
"If you've got an officer who's climbing up the ladder there, and he's got a number of complaints against him, then somebody upstairs should be watching and making some evaluations," Anderson said.
Police Commissioner Charles Ramsey told the Daily News last week that he couldn't deny a cop a promotion based on allegations.
For reasons that are unknown, Ramsey canceled a meeting that had been scheduled Wednesday afternoon with the department's top brass.
When asked for Mayor Nutter's reaction to the People Paper's story about Washington - and to a recent spate of articles about other commanders who've been accused of wrongdoing - mayoral spokesman Mark McDonald wrote in an email: "As you know, Commissioner Ramsey has beefed up resources for the Internal Affairs division and he's committed to rooting out wrongdoing wherever it may be.
"He has challenged police officers to maintain high standards of behavior. He is doing an excellent job in improving the professionalism and capabilities of all members of the department.
"We do not comment on pending litigation as a matter of policy."
Contact David Gambacorta at email@example.com or 215-854-5994. Follow him on Twitter at @dgambacorta.