The NANOGrav pulsar timing array experiment reported evidence for a stochastic common-spectrum process affecting pulsar timing residuals in its 12.5-year dataset, which might be interpreted as the first detection of a stochastic gravitational wave background (SGWB). I examine whether the NANOGrav signal might be explained by an inflationary SGWB, focusing on the implications for the tensor spectral index $n_T$ and the tensor-to-scalar ratio $r$. Explaining NANOGrav while complying with upper limits on $r$ from BICEP2/Keck Array and Planck requires $r gtrsim {cal O}(10^{-6})$ in conjunction with an extremely blue tensor spectrum, $0.7 lesssim n_T lesssim 1.3$. After discussing models which can realize such a blue spectrum, I show that this region of parameter space can be brought in agreement with Big Bang Nucleosynthesis constraints for a sufficiently low reheating scale, $T_{rm rh} lesssim 100,{rm GeV}-1,{rm TeV}$. With the important caveat of having assumed a power-law parametrization for the primordial tensor spectrum, an inflationary interpretation of the NANOGrav signal is therefore not excluded.
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