The letter to the US Food and Drug Administration, first reported by The Associated Press, stopped short of suggesting Ohio is moving forward to obtain the powerful anesthetic sodium thiopental.
However, the state asked to begin discussing with federal officials about acquiring the substance legally.
The FDA had warned Ohio in June that importing the restricted drug could be illegal as a result of recent federal court decisions, setting up the latest roadblock that Ohio and several other states have faced in carrying out the death penalty.
Ohio hasn't executed anyone since January 2014, when condemned killer Dennis McGuire gasped and snorted repeatedly during a 26-minute procedure with a two-drug method that had yet to be tried.
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Ohio abandoned that method in favor of other drugs it now can't find.
Pharmaceutical companies discontinued the medications traditionally used by states in executions or put them off limits for use in lethal injections.
Stephen Gray, chief counsel for the Ohio Department of Rehabilitation & Correction, said the state has no intention of violating the law to obtain such drugs but "the responsibility to carry out lawful and humane executions when called upon by the courts to do so is enormous, and it is a responsibility that ODRC does not take lightly."
Supporters of capital punishment encourage states to continue to pursue legal avenues for getting the drugs or find alternatives so that condemned killers can be brought to justice.
Ohio's latest correspondence comes as the state is set to resume executions in a little over three months.
The state is scheduled to execute Ronald Phillips on Jan. 21 for raping and killing his girlfriend's 3-year-old daughter in 1993. Another 23 executions have been scheduled into 2019.
That was about two weeks after its governor confirmed the state had obtained sodium thiopental from India.
FDA warnings to Ohio and Nebraska followed a federal court ruling two years ago in a case brought by death row inmates in Tennessee, Arizona and California that found the agency was wrong to allow sodium thiopental to be imported for use in executions.