"I have asked the state legislature to pass a very well-thought-out process that respects the Second Amendment and respects due process of law," he said.
"When you have someone who is a threat to themselves or a threat to others, and who has guns, there ought to be a procedure where family members could go into court or police could go into court and present the evidence and have those guns temporarily taken away from that person."
DeWine said such a policy would go a long way to prevent suicides and save lives in general.
He added during Ohio's last budget proceedings, he had asked the legislature in Columbus to put an emphasis on mental health reform, especially in the school system.
"We really have an epidemic," he said.
Of Trump's visit to Dayton, DeWine said the president and first lady were gracious and consoling to victims and appreciative of law enforcement.
"I can just tell you in every room he went, there were family members and sometimes victims and they were all very happy to see the president," he said.
He added he asked the president about passing "red flag" laws, and that Trump, in turn, asked him to forward him a "summary" of his ideas in the near future.
Nationwide, Vermont, Massachusetts, Rhode Island, Connecticut, New York, New Jersey, Delaware, Maryland, Hawaii, Florida, Indiana, Illinois, California, Oregon, Washington, Nevada, Colorado and the District of Columbia have enacted state "red flag" laws.