Alain Connes being a leading French mathematician today one could ask whether he is a member of the Bourbaki group. Is there a published reference that would either refute or confirm this?

Connes was a member, according to M. Mashaal "Bourbaki: A secret society of mathematicians", AMS 2006 (translated from the French by A. Pierrehumbert). It says so on page 18; see the link. But, as was also mentioned there, there is a rule that members retire at age 50, so by that rule he is no longer a member.

pastBourbaki membership is also not kept private AFAIK, so in contrast to some other recent question there is not even this problem, and in any case it asks for a reference. $\endgroup$ – user9072 Mar 13 '16 at 19:26