That would make being caught with a small amount of marijuana the legal equivalent of talking on a cellphone while driving.
Kenney, a potential mayoral candidate next year, said his goal is to stop wasting police and court time on a low-level offense. Last year, 4,200 people were arrested in the city for marijuana possession, he said.
Recent studies by the American Civil Liberties Union have found a racial component to the law as well: More than 82 percent of those arrested in the city for marijuana were black.
"You ask some 20-year-old to empty his pockets because you're stopping and frisking him, and he has two joints on him, I don't think he should be locked up for that," Kenney said.
A number of cities, including Chicago and Washington, have made similar changes.
Kenney's original idea, introduced this year, called for police to issue summonses to offenders rather than taking them into custody. Court and administration officials, however, said the system wasn't equipped to have summonses issued on the street or mailed.
They also said the change wouldn't save police time, since all the information gathered on offenders still had to be entered into the court system, and the drugs still had to be weighed, processed, and tested.
With his new bill, Kenney said, "the whole problem goes away."
"There's no criminal booking," he said. "There's no arrest."
Either of Kenney's bills would allow police to make arrests under a variety of circumstances, including if offenders cannot prove their identity.
Mark McDonald, a spokesman for Mayor Nutter, said the administration would comment on the bill when it is heard by a committee.
Kenney said he intended to call his bill for a final vote before Council's summer recess begins in June.
If opposition extends beyond the administrative hassle, Kenney said, then his new bill will "smoke it out, so to speak."