The deal could be valued at over Rs 500 crore, they said, adding that the exclusivity period of the due diligence is likely to end in next few days.
A Bombay Dyeing spokesperson declined to comment on the queries about stake sale talks, terming them as speculations.
Banking sources said, however, that the PE firms eyeing the stake include big global names like WL Ross, Sequoia Capital and JC Flowers.
Phone calls and text messages to Jeh Wadia, Managing Director of Bombay Dyeing and one of the two sons of the group head Nusli Wadia, did not elicit any response.
The Bombay Dyeing spokesman said in an email that "the company does not wish to comment on baseless and false rumours".
Sources said, however, that the due diligence process is in advanced stage and Wadias plan to use the sale proceeds for its realty arm Bombay Realty, which has a couple of posh projects underway and owns a huge land-bank in Mumbai.
Promoters currently own about 52.3 per cent stake in Bombay Dyeing, while domestic institutions and foreign funds together have about 20 per cent stake.
In FY'13, the company had a revenue of Rs 2,502 crore and a net income of Rs 76 crore.
WL Ross has been bullish on the country for quite some time and has its pie in various sectors. It has made huge profits from a Rs 350 crore investment into low-cost carrier SpiceJet in the past. Sequoia is also heavily invested into the country.
The New York-based JC Flowers have billions in investments and was set in 2001 by a former Goldman Sachs partner J Christopher Flowers.
The diversified Wadia group is one of the oldest conglomerates in the country and was set up in 1736. It is present in businesses like textiles, food, aviation, real estate and trading, among others through entities like Bombay Dyeing, Britannia, GoAir and Bombay Realty.