The aim is “to balance the need to increase efficiency” and fulfil the Government’s plans for a “rehabilitation revolution”, the Ministry of Justice (MOJ) said.
The four Yorkshire prisons concerned are: Lindholme, near Doncaster; Moorland, near Hatfield Woodhouse; Hatfield, near Doncaster, and The Wolds, near Brough in East Yorkshire.
They were selected based on “the potential for efficiency improvements, service reform and innovation, not on the basis of poor performance”, the MoJ said.
The Wolds is already run by security firm G4S and is coming to the end of its current contract, while the others are all being put out for competition for the first time.
A further 2,500 prison places are also expected to become available over the next year.
Mr Clarke said: “The public have a right to expect continuing improvement in the quality and efficiency of public services, without compromising public safety.