Long Island Sen. Dean G. Skelos, the majority leader, said the proposal is too broad and would condone people carrying around marijuana, The New York Times reported.
The proposal would reduce the open possession of 25 grams or less of marijuana from a misdemeanor to a violation.
"Being able to just walk around with 10 joints in each ear, and it only be a violation, I think that's wrong," Skelos told reporters.
Civil rights groups say tens of thousands of people have been arrested in New York City following police stops requiring young people to empty their pockets. Civil rights groups say people have been charged with the misdemeanor when the marijuana is in public view only because of a police stop.
"I think we can work on that," Skelos said. "That is wrong. It should be a violation. You're following the policeman's order."
Josh Vlasto, a spokesman for Cuomo, said the governor would like to talk to the Senate.
"We look forward to working these issues through with the Senate in order to end an injustice that has been allowed to go on for too long," Vlasto said.