Sky News understands that Cinven, whose previous investments have included Fitness First and Pizza Express, has emerged as the frontrunner to acquire the fashion brand.
Cinven is said to have edged ahead of rival bidders including Lion Capital, a former owner of Jimmy Choo, although sources said on Friday evening that a deal was not yet close to being signed.
Kurt Geiger's management team and its backers at Sycamore Partners, an American private equity firm, hired bankers at Goldman Sachs earlier this year to explore a sale of part or all of the business.
An outright sale would be the latest in a string of deals involving the brand, which prides itself on its celebrity customer base and which operates footwear concessions in department stores including Harrods and Selfridges.


Jon Hamm, the Mad Men actor, and ‎the model Rosie Huntington-Whiteley are among prominent fans of the brand.
Kurt Geiger was bought by Jones Group, a US fashion company, in 2011 in a deal worth £215m.
Jones' subsequent troubles led it in April last year into the arms of Sycamore, which days later sold a stake to Kurt Geiger chief executive Neil Clifford and other senior managers.
At the time, Mr Clifford said: "We believe our company has tremendous potential for growth in the UK and internationally, and we will continue to invest in new opportunities alongside our department store and brand partners."
In an interview earlier this year, Mr Clifford indicated that Kurt Geiger could abandon its US operations, citing the need for substantial infrastructure investment to make a success of the move.
In addition to its hundreds of concession operations, Kurt Geiger sells its own brands‎ such as Carvela in more than 70 stores around the world.


Kurt Geiger's first shop opened on London's Bond Street in 1963, but it took until 2007 before its first significant overseas expansion into Europe and the Middle East.
Four years after that, the business was sold to Jones Group by Graphite Capital, a private equity firm which had bought it from Barclays' buyout arm in 2008.
Prior to Barclays, Kurt Geiger had been owned by Mohamed Fayed, the then owner of Harrods.
Cinven and Kurt Geiger both declined to comment.