Bradford Teaching Hospitals NHS Foundation Trust
The trust has a private patient unit called The York Suite  (which has 12 beds):
The York Suite is a dedicated private in-patient ward located within the Duke of York Wing of the Bradford Royal Infirmary. The York Suite combines the advantages of private healthcare with the safety and peace of mind of having all the high technology support and experienced staff of a modern NHS hospital.
The unit is managed by the Bradford Teaching Hospitals NHS Foundation Trust which enjoys the well respected position of having one of the lowest infection rates and being independently rated as one of the safest hospitals in the UK. The York Suite is comprised of single rooms with en-suite accomodation and has recently been re-furbished. It is able to offer a wide range of medical and surgical treatments, under the care of experienced consultants for patients that have medical insurance and those that choose to pay for their care.
The trust indicates that it is already working with private providers :
We have strong relationships with the local Independent Sector Treatment Centre and other private providers.