The Cincinnati Bengals wanted to sign Michael Vick in 2010 after he was released from prison for running an illegal dog fighting ring (among other crimes), but he ultimately chose to sign with the Philadelphia Eagles, who were a much more stable situation or him to go into and rehabilitate his image.
The Bengals went on to eventually replace Carson Palmer with Andy Dalton, and they appear to be committed to him as their franchise QB. Whether he is a true franchise QB remains to be seen, so there's been speculation the team may show interest in Vick once again with him set to hit free agency.
It doesn't appear that will happen, according to Mike Florio of Pro Football Talk:
One place where he won't be competing for the job is Cincinnati. Per a league source, the Bengals have no interest in Vick.
Five years ago, they did. And Vick had interest in them. In 2011, Vick explained that he believed the Bengals and Bills were better fits than the Eagles after Vick returned to the NFL two years earlier following a prison term for dogfighting. Some believe current Bengals starter Andy Dalton isn't the man to get the team beyond the wild-card round of the playoffs, which would make Vick an intriguing, one-year, all-in option.
Regardless, the Bengals don't want to shake Dalton's faith by injecting Vick into the mix.
It does make sense the team doesn't want to "shake" Dalton's confidence. On the other hand, he hasn't had anyone push him for his job since coming to the NFL, and that could lead to him regressing going forward.
The Bengals need to bring a vet in to put a little pressure on Dalton, but it doesn't appear Vick will be one to do so, at least for now.