Brailsford established Team Sky with the aim of producing a first British winner and has succeeded, with Sir Bradley Wiggins' 2012 triumph in the 99th edition followed last year by Chris Froome, the defending champion in this year's race.
Now Brailsford has told the French newspaper L'Equipe of his desire to give France a first winner since Bernard Hinault won the last of his five titles 29 years ago. He said: "We have won the Tour with a British rider, but when are we going to see a French rider win the Tour? That would be enormous."
Asked if that could be his next objective, the former British Cycling performance director added: "Yes, I would like to win with a French rider. I think it needs to happen. For the Tour, for France, for the French, for the sport, having a French winner would be massive. I think about it often. France deserves a French winner."
However, speaking at Horse Guards Parade after the conclusion of yesterday's third stage, Brailsford qualified his statement and made his priorities clear.
"My focus is on doing everything I can to support Team Sky riders to be on the podium and to help Chris to win this race, to inspire people in this country to get involved and to keep on cycling. The comment was more about a national event; if it's won by the same nation's rider, what a fantastic thing that can be. We've seen it with Andy Murray. It would be an exciting thing to see."
Team Sky were established as a British team, in partnership with British Cycling, but just two Britons were selected in Team Sky's nine-rider squad for the 101st Tour. Many of Team Sky's rivals select riders predominantly from their country of origin but Brailsford says riders would not be selected based on their passports.
"I've worked for 15 years to try to put British Cycling on the map and I hope to continue to do that," said Brailsford added. "It's a British team with a British heart, but the team has become more global, there's no doubt about it."
Asked if Team Sky's leader will always be British, Brailsford added: "There are only five or six guys in the world who can win the Tour. Ultimately we're British, but we're performance-based."