GHD, which supplies hair care products to all manner of celebrities, from Madonna and Gwyneth Paltrow to Katy Perry and Jennifer Aniston, made £33.7m in pre-tax profit in its 2015-16 financial year, making a £400m price tag around 13 times earnings.
Sales jumped 9.5 per cent last year to £181.2m.
Lion Capital has appointed bankers at Rothschild to oversee an auction of GHD, Sky News reported.
It's thought other private equity groups are likely to examine bids for GHD, and it was reported documents have been sent to prospective buyers outlining the rapid growth of the company.
Lion picked up GHD from Montagu Private Equity for £300m in 2013.
Montagu bought the company in 2007 for £160m, and doubled profits to more than £32m on revenues of £150m.
Lion Capital planned to grow the business internationally when it acquired it, particularly in the US and Europe.
GHD was launched by three entrepreneurs – Martin Penny, Gary Douglas and Robert Powls – with £15,000 in 2001 from Bradford, West Yorkshire.
The company acquired the rights to a Korean-developed hair-straightening iron, revolutionising the hair straightening process, which could previously only be achieved through salon chemical treatments.
Tangle Teezer, a rival to GHD that was rejected by entrepreneurs on the BBC TV programme Dragon's Den, was last month reported to be in talks with a number of private equity firms.
Lion Capital could not be reached for comment.