The Department for Culture, Media and Sport (DCMS) is considering applications for the post, which was advertised in March and falls vacant in July when the current deputy, Lord Cox, leaves the board of governors.
A spokeswoman said the department hopes to make an announcement soon. She refused to comment on names being considered.
Sources suggest, however, that Lord Puttnam is the favoured candidate in a shortlist of five drawn up from more than 400 applicants.
Lord Puttnam yesterday confirmed he had applied for the job, saying: "I have thrown my hat into the ring... It's a fantastically important job."
In a first for the BBC, current chairman Sir Christopher Bland is involved in the recruitment process for the man or woman who might become his successor. The identities of the rest of the appointment panel, selected by the DCMS, are secret.
Lord Puttnam, producer of the Oscar-laden film, Chariots of Fire, was tipped to be appointed as BBC deputy chairman following his support for New Labour during the last election and his work at the helm of the task force for the arts and culture industries. When the post was advertised, however, reports suggested he was no longer in the running.
Earlier this year Culture Secretary Chris Smith appointed Granada Group chairman Gerry Robinson chairman of the Arts Council. He also appointed lawyer Vanni Treves as chairman of Channel 4 and EMI chairman Colin Southgate as chairman of the Royal Opera House.