We demonstrate that single and binary star clusters can be formed during cloud-cloud collisions triggered by the tidal interaction between the Large and Small Magellanic clouds. We run two different sets of self-consistent numerical simulations which show that compact, bound star clusters can be formed within the centers of two colliding clouds due to strong gaseous shocks, compression, and dissipation, providing the clouds have moderately large relative velocities (10 — 60 km s-1). The impact parameter determines whether the two colliding clouds become a single or a binary cluster. The star formation efficiency in the colliding clouds is dependent upon the initial ratio of the relative velocity of the clouds to the sound speed of the gas. Based on these results, we discuss the observed larger fraction of binary clusters, and star clusters with high ellipticity, in the Magellanic clouds.