Who could succeed Woodhead?
But they are becoming increasingly frustrated by their search for someone who could play a conciliatory role without being too identified with teachers' interests.
Although there are no plans to sack Mr Woodhead - and no action will be taken against him following the allegation that he had an affair with a pupil while she was still at school - he is expected to be eased out before the end of his second term in 2003.
Ministers say he had a useful role when first appointed but is now seen as too provocative to take the education agenda forward.
Mr Woodhead was severely criticised this week by the all-party education select committee who urged him to steer clear of intemperate statements.
Ministers want a chief inspector who can help get teachers on side. One name suggested, but then apparently discarded was Peter Mortimore, director of the Institute of Education, University of London.
Professor Mortimore is well-regarded by Tessa Blackstone, the further and higher education minister. He has been a consistent and vocal critic of OFSTED. He has also questioned some of Labour's education policies, especially plans to measure rising standards in schools. Ironically, the previous Tory government also ruled him out of a top university job because they did not like his views.
Another name being canvassed is Professor David Hargreaves, the vice-chair of the Government's standards task force. But he is seen as being too close to the educational establishment.
Ministers are now looking outside the education world, into the business community. They want to end the polarisation between pro and anti-teacher lobbies.
Select committee report, 6