Bouygues shares rose 4.25 per cent to 33.02 euros and Orange's 1.02 per cent to 12.4 euros following reports that they were discussing the sale.
Business daily Les Echos reported on Thursday that Orange and Bouygues bosses Stéphane Richard and Martin Bouygues "had talks on several occasions", adding that the French government was informed".
But French Prime Minister Manuel Valls told Europe 1 radio on Friday that he "would not comment on rumours, particularly because the two companies were quoted on the stock market".
France's telecoms war seems to be continuing, two months after Vivendi decided to sell SFR to Numericable.
The French government does not seem to hold a common position about the "rumour" as French Economy Minister, Arnaud Montebourg, confirmed the talks had taken place on BFMTV.
Montebourg even expressed an opinion, saying that "The government welcomes and encourages a return to only three operators."
The government holds a 27 per cent stake in Orange.
The turmoil in Bouygues Telecom comes as deals and consolidations flourish across the French telecoms market.
Bouygues recently lost out to Vivendi in a bid for French operator SFR.
The French local market has been plunged into a price war led by low-cost operator Free.
Free, helped by an agreement with Orange, was able to weaken Bouygues and SFR to the point that both looked for partners.
On Monday trade unions at Bouygues expressed the fear that the company might shed 2,000 jobs in a plan to save 300 million euros, which should no longer be a threat if Bouygues sells its telephone subsidiary to Orange.
Meanwhile in the French telecoms war, Numericable announced on Friday morning, new talks to take over Virgin Mobile.