A rival bid has also made by cable operator Numericable, for approximately €14.75 billion.
The offer made by Bouygues includes €10.5 billion in cash and ownership of 46% of the new company after a planned spin-off, which values SFR at €14.5 billion before synergies.
However, as the third-largest operator in France, regulators are concerned that the acquisition of SFR would create a dominant player in the country.
"We will do what it takes for the market to stay competitive," said Philippe Marien, CFO at Bouygues.
The operator announced that talks had not yet begun with French anti-trust regulators, but that the group was ready to sell mobile spectrum and mobile antennas to address any regulatory concerns.
Earlier last year, French telco Vivendi had planned to spin off SFR as the company looked for ways to reduce debt.
SFR and Bouygues signed a deal in January this year to share their mobile networks, in an effort to reduce costs for both companies.
