NFL Network first reported the news Monday.
Reid's visit with Cincinnati is his first known visit of the free-agency period. He became an unrestricted free agent last month after spending the first five seasons of his career with the San Francisco 49ers.
Reid tweeted at the beginning of the league year that he wasn't receiving interest from teams because of his decision to protest during the national anthem.
"The notion that I can be a great signing for your team for cheap, not because of my skill set but because I've protested systemic oppression, is ludicrous," he tweeted. "If you think is, then your mindset is part of the problem too."
Reid, 26, was the first player to kneel next to Colin Kaepernick during the national anthem in protest of racial inequality and systematic oppression against minorities and has remained outspoken regarding his views on the subject.
But Reid said in March that he would not kneel during the national anthem during the 2018 season, but would instead find other ways to take action.
"I'm not saying I'm going to stop being active because I won't," Reid said during Stanford's Pro Day, which he attended in support of his brother Justin, an NFL draft prospect. "I'm just going to consider different ways to be active, different ways to bring awareness to the issues of this country and improve on the issues happening in this country.
"I don't think it will be in the form of protesting during the anthem. I say 'during' because it's crazy that the narrative changed to we were 'protesting the anthem,' and that wasn't the case."
None of the Bengals players have knelt during the national anthem, but they did link arms as a team in a show of unity during the anthem prior to their game against the Packers on Sept. 24. Bengals coach Marvin Lewis has said he is proud of how his team handled things last year, but has also said he doesn't want anything to get in the way of football.
"Our guys have been excellent in this situation," Lewis told reporters at the owners' meetings. in March "Because we are about playing football. They have other agendas, this is not the place to be. On Sunday for us and throughout the week in the building, it's about football. That's how I've approached it."
The Bengals have never shied away from signing controversial players, including cornerback Adam "Pacman" Jones, whose off-the-field issues have followed him throughout his career, and running back Joe Mixon, who fell to the second round of last year's draft after he punched a woman in the face in 2014.
Reid was a first-round pick by the 49ers in 2013 and made the Pro Bowl as a rookie. He has been a strong player throughout his career, playing strong safety, free safety and even some linebacker last season when San Francisco was shorthanded at the position.
He has 10 interceptions and 36 passes defensed in 70 career games.