He is proposing that offenders caught with less than an ounce of weed, like Fleming, should receive a court summons and be let go.
"I don't believe 4,200 people in the city should be arrested every year, taking 17,000 police hours off the street, processing marijuana arrests," Kenney said. "To me, it makes no sense."
His bill was amended Monday to delineate circumstances where offenders could be arrested - including if they cannot prove their identities, have warrants pending against them, or have ignored previous summonses.
Fleming is white, which made him something of an anomaly. The ACLU's studies in recent years have found that more than 83 percent of marijuana arrests in the city were of African Americans.
Kenney noted that in the police district covering the largely white neighborhoods of Roxborough and Manayunk, only one person has been arrested in the last three years on pot charges - and that was a black male.
"You can't tell me that Latinos and white kids aren't smoking reefer, too," Kenney said. "They're just not getting stopped and arrested for it."
His bill, which won unanimous support from the Law and Government Committee on Monday, is not necessary for police to end custodial arrests - the department has that discretion now, Kenney said. But, he said, "we need, as a council, to make a comment or put our stamp of approval" on the idea.
Francis Healy, a special adviser to Police Commissioner Charles H. Ramsey, said the department "supports the basic premise" that "custodial arrests should not be required" for pot possession.
Healy said existing rules could be changed, but only if the police, the courts, and District Attorney Seth Williams are in agreement.
Williams is embarked in that direction. His office has started a diversionary program for offenders caught with small amounts of marijuana, a move aimed at clearing 3,000 cases out of the main court system.
Kenney, whose bill could go to a floor vote this month, said he would not have called for ending pot arrests if the district attorney had not taken that first step.
Williams, in a tepid, two-sentence statement Monday, said he looked forward to working with Kenney "to ensure his proposal can be implemented in a manner that conforms with existing state law."
Last year, Chicago approved a bill similar to Kenney's, and last week the city council in Washington approved a bill that would make possession of small amounts of pot a civil offense.
"The trend has been toward legalization," Kenney said. "We've just got to be sensible about how we enforce the law and treat our people."