The Nike-branded strips are huge money-spinners. France's official shirt costs 85 euros, even for a child, according to the French Football Federation (FFF) website.
"It's dramatic. Stocks have run out all over the world. We can't meet demand. We're ahead of Brazil in numbers of jerseys sold," Noel Le Graet, head of the FFF, told BFM TV in an interview.
"As soon as a jersey turns up in any of our shops across the world, it's snapped up."
Nike, which only got heavily involved in soccer when the World Cup was played in the United States in 1994, is supplying shirts for 10 countries, including Brazil, France and England.
Rival Adidas is sponsoring 12 of the 32 teams, including strong contenders Germany and Spain, along with hosts Russia.
Le Graet said French team players under Didier Deschamps would each get a 150,000 euro (131,560.13 pounds) bonus if "Les Bleus" reach the semi-finals and 280,000 euros for the final.
(Reporting By Brian Love; Editing by Richard Lough and Elaine Hardcastle)
