A longtime starter for FC Nantes during the club's most successful period during the 1970s and 1980s, Bossis was noted chiefly as a left fullback but filled various roles in defense. Bossis spent much of his time at right back during Nantes' championship-winning seasons in 1977 and 1980, in which Thierry Tusseau normally started on the left, but made the left back position his own beginning in 1981. Bossis helped Nantes to finish first or second in every season between 1976 and 1981. The club added a third title in 1983, finishing ten points ahead of second-place Girondins Bordeaux. In 1985, Bossis moved to the ambitious RC Paris, but the Parisian club achieved only modest success in spite of heavy spending that acquired such players as Enzo Francescoli and Pierre Littbarski. Bossis returned to Nantes for one final season in 1990, lining up next to future French international Marcel Desailly before retiring from play.
Bossis also represented France for ten years, appearing at the World Cups of 1978, 1982, and 1986. He is mostly remembered for missing the last penalty in 1982 World Cup semifinal between West Germany and France. While the score was tied at 4-4, Bossis missed the next penalty, allowing Horst Hrubesch to score the last penalty and drive the Germans to the final. From 1985 to 1992, he held the French record of caps, before fellow defender Manuel Amoros established a new mark with 82 caps. He also held the French record of matches played in the FIFA World Cup with 15, subsequently surpassed by Fabien Barthez. Bossis was also in the team that won the European Championship in 1984.
His younger brother, Joël, also played professional football and holds the all-time record most goals scored for Chamois Niortais.