A system of interdependent networks was recently found to be very vulnerable since cascading failures that may lead to abrupt breakdown of the system. We develop an analytical method, based on the percolation method developed for single networks [M.E.J. Newman, Phys. Rev. Lett. {\bf 103}, 058701 (2009)], to study the effect of clustering within the networks on the robustness of the interdependent networks. We find that, in contrast to single networks where the percolation threshold, $p_c$, does not change with clustering for site percolation and {\it decreases} with clustering for bond percolation, $p_c$ for interdependent networks {\it increases} when networks are more clustered.