It is true if the projective dimension of $M$ over $A$ is at most $3$, and counter examples exist when the projective dimension is $4$.

The first counter example was given in Lucho Avramov's paper "Obstructions to the existence of a multiplicative structure on minimal resolutions". A simplified example, ($A=k[x_1,x_2,x_3,x_4], M= A/(x_1^2, x_1x_2, x_2x_3, x_3x_4, x_4^2)$) together with discussion of related and more recent results can be found in Section 2 of this note (available on Avramov's website).

It is true if the projective dimension of $M$ over $A$ is at most $3$, and counter examples exist when the projective dimension is $4$.

The first counter example was given in Lucho Avramov's paper "Obstructions to the existence of a multiplicative structure on minimal resolutions". A simplified example, ($A=k[t_1,t_2,t_3,t_4], M= A/(t_1^2, t_1t_2, t_2t_3, t_3t_4, t_4^2)$, here you can choose the $x_i$s to be any regular sequence in the annihilator of $M$) together with discussion of related and more recent results can be found in Section 2 of this note (available on Avramov's website).