Davis had been in negotiations with the commercial broadcaster and was tipped as favourite to take over from current executive chairman Michael Grade.
Bishop stepped down as chairman of BMI British Midland in July.
The broadcaster is searching for a chairman and chief executive to take over from Grade who currently holds both roles.
Negotiations with former BSkyB chief Tony Ball to take over as ITV chief executive collapsed at the end of September. At the same time Grade announced that he would also step down as chairman.
ITV has said that the search to fill the chief executive position will resume once a new chairman has been appointed.