The only rumors I have heard about Cincinnati are that they were interested in McDaniels (he is not interviewing), Vance Joseph, and Hue Jackson. Supposedly, Hue is telling people that he's the front runner in the clubhouse for the job and looking at staff options.
There is a big joke in here about Gruden signing Peterman, the Bengals possibly hiring Jackson, and Miami's owner coming out and saying "You know, going 3-13 ain't that bad."
Gase was a red hot commodity when we hired him. He was actually so highly regarded that he straight up turned down a head coaching offer because he didn't like one of the stipulations on it, and elected to spend another year or so as a coordinator. He still has some of that aura, and he still has a great agent (Jimmy Sexton, IIRC), but I think other clubs are (rightly) concluding that he was at least some substantial part of the reason why the Miami Dolphins imploded under his watch.
I think Gase will get another shot as a head coach. If not in New York this season, he will rehabilitate his image somewhere as a coordinator and get another shot. He still has major clout with some media guys, a strong agent, and some residual mojo from the Peyton Manning anointment.