It heralds a major shake-up for soccer in France, putting vast wealth behind a potential challenger to the dominant Ligue 1 team of recent years, Qatar-funded Paris Saint-Germain.
A statement from the Arnault family's holding company, Agache, said they will become the club's majority shareholder. Energy drink giant Red Bull will come aboard with a minority stake.
The statement did not give a monetary figure for the deal, which remains subject to completing legal and other paperwork.
But the billionaire family's company intends to provide the club "with the necessary resources" and wants "to permanently establish the men's and women's teams among the elite of French football and within the hearts of the Parisians".
More broadly, the takeover of the club with a distinctive blue-and-white Eiffel Tower logo could help the French capital build on the sporting momentum of the Paris Olympics and put an end to its place as something of an oddball in the European soccer landscape.
Despite being a powerhouse of fashion, finance, luxury and entertainment, Paris trails far behind London, Madrid and other cities by having just one top-flight soccer team: The hegemonic PSG.
The former club of superstar Kylian Mbappé are a 12-time champion of Ligue 1, with 10 of those titles in France's top league coming after gas-rich Qatar began pouring their wealth into the club they purchased in 2011.
London, by contrast, has seven Premier League clubs this season. Madrid and its suburbs have five clubs in La Liga. Rome has two, sharing Stadio Olimpico.
The Arnault family are expected to initially take a 55 per cent stake and Red Bull around 15 per cent, with current owner Pierre Ferracci keeping the rest for now.
"The history and the evolution of Paris FC embodies a whole other aspect to football in the capital. With the arrival of Agache as the club's majority shareholder, the club will take on a new dimension with new goals and criteria for success," the family company's statement said.
Created in 1969, Paris FC's men's team has yet to achieve any significant success but currently top the second-division standings. The women's team already play in the top-tier Première League.
Red Bull will mainly act as a sporting adviser, "whether it be reinforcing the detection of young talent capable of joining the training centre or targeting the best players".
Red Bull this month announced former Liverpool coach Jürgen Klopp will become their head of global soccer from January, overseeing its international network of clubs.
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