If successful the deal would have combined the current No. 2 and No. 3 mobile operators to create France's biggest telecoms group and taken the French mobile market from four to three players at a time when the merits of such consolidation are being hotly debated in Europe.
The board's decision also follows opposition from President Francois Hollande's Socialist government who had expressed concern over the deal, saying it could be bad for jobs, consumers and investment.
Bouygues's board said it paid "great attention" to the social consequences of the deal.
Martin Bouygues, scion of one of France's top industrialist families, decided against parting with the telecoms business that is his main contribution to the conglomerate built by his father.
The 63-year-old Bouygues has already rebuffed at least two offers in the past year for the unit he founded in 1994, insisting as recently as last month that Bouygues Telecom could prosper on its own. He once replied to a question about selling the business: "And you, would you sell your wife?"
But the offer by Drahi outstripped past approaches. Drahi's own empire-building took off in 2014 when he beat out Bouygues to acquire Vivendi's SFR, the second-biggest mobile operator.
The 10 billion-euro price tag would have valued the unit as much as the entire Bouygues group before the offer was made public. It also eclipsed earlier offers from Orange and low-cost operator Iliad - each around 5.5 billion euros - as well as a previous 8 billion-euro approach by Drahi.
The business, which has been posting losses since Iliad broke into the French mobile market three years ago, had a 2014 total asset value of 5.8 billion euros.
The board said Bouygues Telecom had the means to return to a 2011 EBITDA margin of at least 25 percent by 2017.