Hayward, who left BP in the wake of the Gulf of Mexico oil spill, has led Kurdistan-focused Genel since 2011.
He founded the company with British financier Nat Rothschild.
While there is no explicit rule against a chief executive being appointed as chairman of the same company, the UK Corporate Governance Code encourages against it and operates a “comply or explain” policy around such hires.
Hayward is expected to be replaced as chief executive by Murat Ozgul, who head up the firm’s operations in Kurdistan, according to a report in the Sunday Times.
A spokesman for Genel said yesterday: “Discussions on management succession are ongoing, and when a decision is made it will be announced to the market.”
He added: “We have previously signalled the likelihood that Tony Hayward may become chairman of Genel, a move key stakeholders have told us they would welcome.”
Genel’s revenues jumped by 49 per cent year on year in 2014, but it swung to a pre-tax loss of $312.8m (£204.7m) after depreciation charges and write offs from exploration wells.