It has been recently shown that the cosmic repulsion can have a highly significant influence on the motion of Magellanic Clouds (MC) in the gravitational field of Milky Way, treated in the framework of the Cold Dark Matter (CDM) halo model. However, there is an alternative to the CDM halo explanation of the rotation curves in the periphery of spiral galaxies, based on MOdified Newtonian Dynamics (MOND). Therefore, we study the role of the cosmic repulsion in the framework of the MOND theory applied to determine the MC motion. Our results demonstrate that in the MOND framework the influence of the cosmic repulsion on the motion of both Small and Large MC is also highly significant, but it is of a different character than in the framework of the CDM halomodel. Moreover, we demonstrate that the MC motion in the framework of the CDM halo and MOND models is subtantially different and can serve as a test of these fundamentally different approaches to the explanation of the phenomena related