The successful applicant will replace Sir Hugh Taylor, who is resigning to take up the role of chairman at Guy's and St Thomas' Foundation Trust in London.
The advert says: 'This is not a job for the faint hearted but calls for an exceptional leader, with credibility, drive, tenacity and the ability to work with partners at all levels and from all sectors.
'This job will be extremely demanding, but also highly rewarding, with a once-in-a-lifetime opportunity to leave an enduring legacy for this and future generations.'
The salary on offer is not stated on the job advertisement. However, a Cabinet Office report recently revealed that Sir Hugh was earning between £155,000 and £159,999.
NHS chief executive David Nicholson was reported to earn £255,000 a year.