ABSTRACT The North American Nanohertz Observatory for Gravitational Waves (NANOGrav) recently reported evidence for the presence of a common stochastic signal across their array of pulsars. The origin of this signal is still unclear. One possibility is that it is due to a stochastic gravitational-wave background (SGWB) in the ∼1–10 nHz frequency region. Taking the NANOGrav observational result at face value, we show that this signal would be fully consistent with an SGWB produced by an unresolved population of in-spiralling massive black hole binaries (MBHBs) predicted by current theoretical models. Considering an astrophysically agnostic model, the MBHB merger rate is loosely constrained. Including additional constraints from galaxy pairing fraction and MBH–bulge scaling relations, we find that the MBHB merger rate is ${1.2\times 10^{-5}}{\rm -}{4.5\times 10^{-4}}\, \mathrm{Mpc}^{-3}\, \mathrm{Gyr}^{-1}$ , the MBHB merger time-scale is $\le 2.7\, \mathrm{Gyr}$, and the norm of the MBH−Mbulge relation is $\ge 1.2\times 10^{8}\, {\rm M}_\odot$ (all quoted at 90 per  cent credible intervals). Regardless of the astrophysical details of MBHB assembly, the NANOGrav result would imply that a sufficiently large population of massive black holes pair up, form binaries and merge within a Hubble time.