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 that show that compact, bound star clusters can be formed within the centers of two colliding clouds as a result of strong gaseous shocks, compression, and dissipation, provided that 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 on 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.