The music mogul recently confirmed that he would be departing the series in order to concentrate on the launch of the US version of The X Factor.
Since then, names including Howard Stern, Noel Gallagher, La Toya Jackson, Rodney Jerkins and Ryan Seacrest have been linked to the judging role.
In an interview with The Hollywood Reporter, Fox president Kevin Reilly said that none of the rumoured contenders will be offered the job.
"There's just all sorts of conjecture out there, but I can tell you there's been some interesting dialogue and interesting names. There's going to be some very interesting developments on that front."
He added: "I just want to say credibility is going to be very important to us. That's going to be important to maintain."
American Idol continues Tuesday at 8pm on Fox.