Mr Clarke has told officials to draw up a list of between 12 and 20 institutions which could be privatised within the lifetime of this Parliament.
One of these seems certain to be Doncaster in South Yorkshire, which will be opened in 1994 as part of the present prison-building programme. This will have a unit containing top-security Category A inmates.
At a recent meeting with senior civil servants, Mr Clarke also floated the idea of allowing prisons to become self-governing trusts, a policy he introduced to the National Health Service as Secretary of State for Health.
So far, the Government has contracted out the management of the Wolds remand prison on Humberside and is inviting tenders for Blakenhurst, a local prison due to open in Hereford and Worcester next year.
Stephen Shaw, director of the Prison Reform Trust, described the new plans as 'far more radical than anything yet imagined. If implemented in full, they would represent a monumental leap in the dark.'
He pointed out that the decision to ask a private company to run the Wolds had been described by ministers as an experiment. 'It appears to be the shortest-run experiment in history,' he said.
Publicly, Mr Clarke has said only that he is considering an extension of the privatisation programme to existing establishments. However, his order to list at least 12 potential candidates - almost 10 per cent of the prison service - shows that the idea has taken root at the Home Office.
It is likely that prison governors could compete for contracts alongside private companies.
Ministers say that their policy of contracting out public services has brought wide-ranging benefits in a number of areas, from local to central government. There is no reason why this should not work for prisons, they believe.
But any move to contract out existing jails would certainly set the Government on a collision course with the Prison Officers' Association, one of the strongest trade unions in the country.
A high proportion of prison officers are members of the POA, who would fiercely resist attempts to rewrite their contracts or replace them with new employees.
Staff would also be wary of moves towards making prisons self-governing trusts, although it is by no means clear what this would entail. Lord Justice Woolf said in his report on the Strangeways riots that prison governors should be given greater autonomy to decide how they ran their jails.
Self-governing status could mean that they are given the power to hire and fire staff and set wage levels - something which would also meet with hostility from prison officers.
'This also calls into question the Government's plans to grant the Prisons Department agency status and move it to a new headquarters in Derby,' Mr Shaw said.