"This will destroy Mitch McConnell in Kentucky," Craig Davis, president of the United Gamefowl Breeders Association, told the newspaper.
The bill made attending a cockfighting event a federal misdemeanor, punishable by up to a year in prison and a $100,000 fine.
Davis, who confronted McConnell with hundreds of other "cockfighting enthusiasts" at a Monday event, said he could move "as many as 60,000 votes" if McConnell doesn't address the issue.
He said that some UGBA members had already proposed putting their support behind McConnell's Republican primary challenger, Matt Bevin, but Davis wanted to give McConnell a chance to right his wrong.
"When you make a law like that you take good taxpaying people and you turn them into criminals overnight," he said. "The grassroots on this are not playing games anymore. They've been beaten and battered for 30 years. They're rural people. They want to be left alone."
McConnell's office told the newspaper that his staff was willing to hear their grievances.