The former minister, who was forced reluctantly to resign his post as Heritage minister in John Major's government, is one of several Conservative candidates being considered as part of an effort to bring political balance to the senior management of the BBC.
The Government is keen that a conservative supporter is appointed as deputy, in an effort to diffuse the controversy surrounding its appointment of the Labour-supporting economist Gavyn Davies last week as the new chairman of the corporation. Davies joined director general, Greg Dyke, who is also a Labour supporter.
Michael Portillo, the former shadow chancellor, has also been mentioned as a possible candidate, but this is seen as more unlikely as he is thought to want to remain an MP.
Mellor, who lost his ministerial job in 1992 after a tabloid exposed his affair with actress Antonia de Sancha, has since worked as a radio presenter on BBC Radio 5 Live and more recently on Classic FM. He would have to give up his presenting duties if he took the deputy chairman post.
Other Tory names mentioned include Michael Heseltine and the economist Baroness Hogg who worked for John Major's government.
The deputy chairman's job is a one day a week post which comes with a £25,000 salary.