Bach never published these, and certainly not as the set they are now. He also didn’t call them French. Suites they are, as they are dances. Baroque suites are that. Dances, in the French fashion in this case. The earliest versions of these that are traceable are from 1722, when they were copied for Bach’s second wife, Anna Magdalena.
When biographer Johann Forkel made an overview of Bach’s works in 1802, he listed these separate compositions as one work, and thus created the French Suites in a way.
The best known of these is perhaps the fifth, in many version played on a piano. In my mind they should be played on the instrument they were written for: a “klavessin“.