The bare propagator

$(p^2+m^2)^{-1}$ for a scalar field, $(\gamma\cdot p+m)^{-1}$ for a Dirac field

is the inverse of the Fourier transformed differential operator inside the full quadratic part of the Lagrangian density. It includes the kinetic term ($p^2$ for a scalar field, $\gamma\cdot p$ for a Dirac field) and the mass term ($m^2$ for a scalar field, $m$ for a Dirac field).

In renormalization theory, the bare mass term is replaced by the physical mass tern and the difference is added to the interaction as a mass counterterm. This is the first step needed to make the perturbation theory finite. In a second step, one adds momentum-dependent self-energy terms to get the renormalized propagator. Only the latter is a physical propagator with observable information.