Barrieu was assistant coach at Le Havre, the club Bradley left to join the Swans on 3 October.
The 44-year-old had previously worked with the USA national team, LA Galaxy and Sheffield Wednesday.
At the time of his appointment, Bradley said he wanted to add one or two coaches and hinted at bringing in someone with Premier League experience.
Three members of the backroom staff - Diego Bortoluzzi, Gabrielle Ambrosetti, Claudio Bordon - left when Guidolin was dismissed.
First team coach Alan Curtis, who has been associated with the club as a player, coach and caretaker manager since the 1970s, is expected to stay on.
"For me Alan Curtis will be very important, he's going to be my older brother - only a little older," said Bradley.
"His sense of the club, his eye, his experience will be important. I'm working on getting the right balance."
Bradley is the first American to manage in the Premier League and takes charge of his first match when Swansea play Arsenal at the Emirates Stadium on Saturday, 15 October.
Swansea manager Bob Bradley is appointing Pierre Barrieu to his backroom staff the Liberty Stadium.