Police have yet to identify the dead man and during a pair of news briefings on Thursday evening declined to comment on his motive.
The New York Times and NBC News, citing unnamed sources, identified him as Ricky Shiffer, 42.
A man going by that name forewarned of the incident on Truth Social, the messaging medium created by former US president Donald Trump.
"If you don't hear from me, it is true I tried attacking the F.B.I., and it'll mean either I was taken off the internet, the F.B.I. got me, or they sent the regular cops ...", the message said.
Posts from the same user name, including some expressing extreme views, were taken down shortly after police told reporters the Cincinnati suspect had been killed.
NBC News, citing two unnamed officials familiar with the matter, said the suspect was at the US Capitol building in Washington during the assault by Trump supporters on January 6, 2021.
The New York Times, citing two law enforcement officials familiar with the matter, reported investigators were looking into whether the suspect had ties to extremist groups.
Some Trump supporters claim a grievance with the FBI over the former president's entanglements with the agency which have included its probe into Russia's support for his 2016 presidential campaign and Monday's court-authorised search of his Florida home.
The FBI has been the subject of online threats since its agents searched Trump's Mar-a-Lago estate as part of an investigation into documents removed from the White House when Trump left office in January 2021.
FBI Director Christopher Wray, who has described homegrown violent extremism as the most significant security threat to the United States, issued a statement following Thursday's events decrying "unfounded attacks on the integrity of the FBI".
"Violence and threats against law enforcement, including the FBI, are dangerous and should be deeply concerning to all Americans," Wray said.
The Ohio incident began early on Thursday, when the suspect unsuccessfully tried to enter the FBI's visitor screening facility. He fled the area in a white vehicle, the FBI said.
A chase ensued and the suspect fired his weapon at a state trooper, Ohio State Highway Patrol spokesman Nathan Dennis told a news briefing.
The vehicle eventually came to a stop in Clinton County and gunfire was exchanged between the suspect and police, he said, before a lengthy standoff that ended with the suspect being shot six hours after the initial incident in Cincinnati.
"Throughout the day today law enforcement officers attempted to negotiate with the suspect. After a time the negotiations failed," Dennis said.
"The suspect then did raise a firearm toward law enforcement and shots were fired by law enforcement officers on the scene," Dennis said.
He said the suspect died at the site and no police were injured.
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