  We study the electron-phonon coupling strength near the interface of
monolayer and bilayer FeSe thin films on SrTiO$_3$, BaTiO$_3$, and
oxygen-vacant SrTiO$_3$ substrates, using ab initio methods. The calculated
total electron-phonon coupling strength $\lambda=0.2\text{--}0.3$ cannot
account for the high $T_c\sim 70$ K observed in these systems through the
conventional phonon-mediated pairing mechanism. In all of these systems,
however, we find that the coupling constant of a polar oxygen branch peaks at
$\mathbf{q}=0$ with negligible coupling elsewhere, while the energy of this
mode coincides with the offset energy of the replica bands measured recently by
angle-resolved photoemission spectroscopy experiments. But the integrated
coupling strength for this mode from our current calculations is still too
small to produce the observed high $T_c$, even through the more efficient
pairing mechanism provided by the forward scattering. We arrive at the same
qualitative conclusion when considering a checkerboard antiferromagnetic
configuration in the Fe layer. In light of the experimental observations of the
replica band feature and the relatively high $T_c$ of FeSe monolayers on polar
substrates, our results point towards a cooperative role for the
electron-phonon interaction, where the cross-interface interaction acts in
conjunction with a purely electronic interaction. We also discuss a few
scenarios where the coupling strength obtained here may be enhanced.
