The move gives the board of House of Fraser, which has Scottish roots, an alternative avenue to the stock market flotation it was expected to pursue next year.
The 164-year-old House of Fraser trades from 61 stores in Britain and Ireland, including Jenners in Edinburgh, and has annual sales of £1.2 billion.
Sir Tom became an 11.1% shareholder in the business following his participation in the Highland consortium's £351 million takeover in the summer of 2006. Backers of the deal included Halifax Bank of Scotland, now part of Lloyds Banking Group.
House of Fraser is expected to attract a valuation of £350m to £450m, suggesting Sir Tom's stake is worth up to £50m.
A spokesman for Sir Tom said of the Galeries Lafayette approach: "It is not something we can comment on."
House of Fraser made a loss before tax and exceptional items of £6.9m in the year to January 26 when it had a debt pile of £157.2m and a pension deficit that stood at £46.9m.
Some 49% percent of House of Fraser's parent Highland Group Holdings is owned by Icelandic banks Landesbanki and Glitnir, who inherited their stakes on the failure of fellow Icelandic institutions Baugur and FL Group during the credit crunch.
The House of Fraser buyout was one of the largest in a series of retail take-overs by Baugur chief Jon Ásgeir Jóhannesson ahead of the credit crunch. Others included toy store Hamleys, food chain Iceland and fashion retailer Oasis.
House of Fraser chairman Don McCarthy who sold his Shoe Studio chain to Baugur owns 20% of the department store chain, senior managers hold 6%. Retail entrepreneur Kevin Stanford, who sold Baugur to his Karen Millen business in 2004, owns 9%, while part-nationalised Lloyds has 5% thanks to its acquisition of HBOS in 2008.
The takeover of House of Fraser would fit with the strategic approach of Galeries Lafayette, which has responded to weak consumer spending in France by expanding abroad. Hitherto this has focused on the rapidly expanding moneyed classes in emerging markets with stores in Jakarta, Beijing and Dubai.
In all it has 65 outlets, with revenue of €2.3bn (£1.9bn) last year.
Independent retail analyst Nick Bubb said a sale to Galeries Lafayette or a stock marketing listing "may be rather fanciful hopes" for a business which he said had a distressed balance sheet and shareholder base.
Earlier this year it was reported that a Qatari sovereign wealth fund had held talks with House of Fraser shareholders.
Sports Direct's Mike Ashley, who owns Newcastle United Football Club and is a business ally of Sir Tom, said last year that he had looked at acquiring a minority shareholding in House of Fraser
Galeries Layayette's interest comes at a tough time for retailers due to fears that the crucial Christmas shopping season has been weak. Shares in House of Fraser's stock market listed rival Debenhams dropped 3.55p, or 4.33%, to 78.45p yesterday. At the start of the year, its shares had tipped 118p.
The events are the latest in a colourful history for House of Fraser, whose first store was opened in 1849 by Hugh Fraser and James Arthur on Glasgow's Buchanan Street.
The firm launched on to the stock exchange in 1948 when it had 16 stores in Scotland and for the next 20 years expanded south of the Border.
The late Sir Hugh Fraser took over the empire from his father Lord Fraser of Allander in 1966 but disclaimed his peerage.
The colourful entrepreneur stepped down from the chairmanship in 1981 after Lonrho attempted a takeover. It was eventually taken private in 1985 by the Al Fayed family for £615m and Fraser died of lung cancer two years later, aged 50.
In 1994, House of Fraser went public for the second time and was bought by the Highland consortium 12 years later.