Chaotic terrain on Europa is interpreted to be the result of the breakup of brittle surface materials over a mobile substrate. Two end-member models have been proposed for the mobile substrate: a liquid water ocean or solid warm ductile ice. Direct contact between the brittle surface ice and an underlying ocean might be accomplished by melting through Europa's ice shell, but plausible sources of thermal energy are insufficient to locally melt through the ice shell. We calculate that a liquid or partially liquid substrate is probably necessary for tilted blocks to be observed, challenging a solid-state model. We propose that a model in which chaos areas on Europa formed over bodies of melt or partial melt within the ice shell may be the best match to observational and thermal constraints. Such bodies of melt within the ice shell may be triggered by warm diapirs of relatively clean ice that partially melted an overlying layer of lower melting temperature, such as ice contaminated with salts.