LVMH, which owns global brands including fashion labels Louis Vuitton and Donna Karan, and champagne Moet Chandon, is reported to be buying a stake in Edun of just under 50%, although no official details of the deal have been disclosed.
Edun, which Bono and his wife launched in 2005, is aimed at promoting fair trade by selling goods made from ethically sourced cotton from developing countries such as India, Kenya and Uganda.
Bono, whose real name is Paul Hewson, said: "Edun is a start-up that could make a splash. With LVMH we can accelerate its development."
Bono has a reputation as an ethical campaigner and was a Nobel Peace Prize nominee.
He is a co-founder of charity brand Red, set up to combat the spread of Aids in Africa.
Bernard Arnault, LVMH's chief executive, said: "We are proud to contribute to the operational development of Edun and to the improvement of living standards of local communities."
Edun's headquarters are in Dublin and it has a subsidiary in New York.
LVMH's interest in the firm will be overseen by Mark Weber, the head of Donna Karan International.