The need to appoint a chairman is pressing given Friday's announcement by the company that not only would Tony Ball not be given the chief executive position, but that executive chairman Michael Grade wanted to leave as soon as possible.
In addition to Davis and Bishop, former BBC and BT chairman Sir Christopher Bland and British Airways chairman Martin Broughton have been approached, according to The Sunday Times.
Davis is seen as the frontrunner and has media experience from several years at the helm of business, legal and scientific publisher Reed Elsevier. He has won praise for moving the company's offering online.
However, leading shareholders in ITV are ready to block Davis from the chairman role, according to The Sunday Telegraph.
The paper quoted one "major shareholder" as saying Davis' time at Reed was a "disaster" and another "top 10 shareholder" as saying: "Following a string of expensive and unnecessary acquisitions, he left the business under-invested and with the highest level of debt in its history."
A response from an ITV spokesman left some doubt as to the extent of the opposition to Davis. The spokesman said: "Obviously shareholders are entitled to their own views. But the board has discussed potential candidates with shareholders who account for half the register and they are supportive of the candidates being considered."
Bishop, the other leading contender to succeed Grade, was chairman of Channel 4 between 1993 and 1997. He also founded airline British Midland, also known as BMI, and sold out to Germany's Lufthansa last year.
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