  The NANOGrav Collaboration has recently published a strong evidence for a
stochastic common-spectrum process that may be interpreted as a stochastic
gravitational wave background. We show that such a signal can be explained by
second-order gravitational waves produced during the formation of primordial
black holes from the collapse of sizeable scalar perturbations generated during
inflation. This possibility has two predictions: $i$) the primordial black
holes may comprise the totality of the dark matter with the dominant
contribution to their mass function falling in the range $(10^{-15}\div
10^{-11}) M_\odot$ and $ii$) the gravitational wave stochastic background will
be seen as well by the LISA experiment.
