At a news conference in Columbus, Kasich said current laws permit state officials to make arrests and confiscate animals if there is evidence of inhumane treatment, The (Newark, Ohio) Advocate reported.
Kasich signed an executive order directing the state Department of Natural Resources to set up a hotline for animal-abuse reports and said Ohio will work with zoos to care for any confiscated animals, the newspaper said.
"We don't know where they all are," Kasich said.
Terry Thompson of Zanesville, who kept exotic animals on his rural property for years, fatally shot himself Tuesday after releasing 56 animals, including bears, lions, tigers, leopards, wolves and monkeys.
Law enforcement officers were forced to shoot and kill 48 of the animals, with others taken to the Columbus Zoo.
Ohio has at least 20 locations similar to the home of the Zanesville man who released his collection of exotic animals and then killed himself, officials said.
"We're talking about the ones we know about," Tim Harrison of Outreach for Animals, an advocacy group that wants to end individual ownership of wild animals, said.
"The sad part about this is these individuals will go unknown until there's a house fire or (an animal) escapes," Harrison told the Zanesville Times-Recorder Friday.
Harrison said he thinks there are more than 20 private operations functioning under the radar and keeping dangerous animals without the state being aware of them.
"The neighbors are the only ones who know about it," he said.