We present a solution to the strong CP problem in which the imaginary component of the up quark mass, $ℐ[m_u]$, acquires a tiny, but nonvanishing value. This is achieved via a Dirac seesaw mechanism, which is also responsible for the generation of the small neutrino masses. Consistency with the observed value of the up quark mass is achieved via instanton contributions arising from QCD-like interactions, as is the case in the closely related massless up-quark solution to the strong $CP$ problem. In our framework, however, the value of the neutron electric dipole moment is directly related to $ℐ[m_u]$, which, due to its common origin with the neutrino masses, implies that the neutron electric dipole moment is likely to be measured in the next round of experiments. We also present a supersymmetric extension of this Dirac seesaw model to stabilize the hierarchy among the scalar mass scales involved in this mechanism.