Vieira, who was born in Senegal's capital Dakar, succeeds Pape Thiaw, whose tenure ended after Senegal's elimination from this year's World Cup in the round of 32.
"Mr Patrick Vieira is the new coach of Senegal's senior national team. The date of the official presentation and the next steps in his installation will be the subject of a subsequent announcement," the FSF said in a statement on X.
The 50-year-old takes charge of a national team for the first time after coaching stints at clubs in England, France and Italy.
Vieira managed Crystal Palace in the Premier League and French sides Nice and Strasbourg, while also spending a brief spell in charge of Serie A club Genoa.
One of France's most decorated midfielders, Vieira was part of the team that won the 1998 World Cup and Euro 2000, earning more than 100 caps for the national side.
At club level, he is best known for his spell with Arsenal, where he made more than 400 appearances and won three Premier League titles.
His first matches in charge of Senegal will come in next month's international window as the 2021 Africa Cup of Nations champions continue preparations for the 2027 tournament. The qualifiers will begin as early as September.