The secretive American hedge-fund investment firm’s interest has sent the likely sale price soaring beyond initial forecasts of £650m.

New York-based Fortress, a key investor in British property manager Mapeley as well as the US retirement-home operator Brookdale Living Communities, is on a shortlist of bidders thought to include private- equity firms BC Partners, Cinven, Charterhouse and Electra.

NM Rothschild, the investment bank, is selling Priory on behalf of its majority shareholder, Doughty Hanson. A deal could eventually realise as much as £750m.

Barchester Healthcare, the group backed by the Irish tycoons John Magnier and JP McManus, is understood to have withdrawn from the auction, as has Macquarie, the acquisitive Australian bank that has been snapping up infrastructure assets across Europe in recent months.

Barchester is, however, understood to be the frontrunner in the auction of Associated Nursing Services, which is also being sold by Rothschild.

The Priory is Britain’s most famous rehab clinic, having treated celebrities such as Ronnie O’Sullivan, the snooker player, and Kerry Katona, the former Atomic Kitten singer.

Apart from the high-profile London site, Priory Healthcare owns 15 psychiatric hospitals, seven teaching schools and several smaller treatment centres.

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