Cloud compression by external shocks is believed to be an important triggering mechanism for gravitational collapse and star formation in the interstellar medium. We have performed MHD simulations to investigate whether the radiative interaction between a shock wave and a small interstellar cloud can induce the conditions for Jeans instability and how the interaction is influenced by magnetic fields of different strengths and orientation. The simulations use the NIRVANA code in three dimensions with anisotropic heat conduction and radiative heating/cooling at an effective resolution of 100 cells per cloud radius. Our cloud has radius 1.5 pc, has density 17 cm(-3), is embedded in a medium of density 0.17 cm(-3), and is struck by a planar Mach 30 shock wave. The simulations produce dense, cold fragments similar to those of Mellema et al. and Fragile et al. We do not find any regions that are Jeans unstable but do record transient cloud density enhancements of factors similar to 10(3)-10(5) for the bulk of the cloud mass, which then decline and converge toward seemingly stable net density enhancement factors similar to 10(2)-10(4). Our run with a weak, initial magnetic field (beta = 10(3)) perpendicular to the shock normal stands out as producing the most lasting density enhancements. We interpret this field strength as being the compromise between weak internal magnetic pressure preventing compression and sufficiently strong magnetic field to thermally insulate the condensations, thus helping them cool radiatively.