Bouygues is internally valued at approximately €8 billion and the discussions have now been left at a standstill.
French regulators have supported the idea of France becoming a three-player market, and there have been rumours in recent weeks that Orange is also interested in a potential tie-up with Bouygues.
Much of the drive towards consolidation in Europe is coming from companies looking to combine operations to counter the costs of providing high-speed 4G networks.
Bloomberg reports that both Iliad and Bouygues have refused to comment on the situation.
Bouygues is seeking a sale of its operations and has also reportedly put on hold plans to share its mobile network with France's second-largest player SFR, with SFR likely to seek compensation if Bouygues pulls out of the agreement as a result of any deal.
Network-sharing agreements are becoming commonplace in Europe to save money on infrastructure, with France at the forefront of this framework.
Following a price war in the country as a result of the entrance of Iliad's low-cost mobile operator Free Mobile, there have been reduced profits and revenues for the country's largest players.
