Let $R$ be a commutative ring, $M$ an $R$-module, $M^*=Hom_R(M,R)$ its dual. What are sufficient (and possibly necessary) conditions on $M$ that ensure that $M^*$ is flat? Is there a name for such modules?

PS I would call such a module *coflat* if this term were not already used for something else.

PPS As $M^\ast$ is clearly torsion-free, I already know, thanks to this beautiful website, some conditions on $R$ that make all $M^*$ flat. I also know about reflexive modules.