Scenario of burrowing of the Co adatoms and clusters on the Cu(001) is considered. Performing atomic scale calculations we find that the Co/Cu interface is stabilized when the Co clusters reside in the surface layer. We demonstrate that coating the Co islands with the Cu substrate material leads to large capillary forces, which promote burrowing. A vacancy mechanism of burrowing is discussed. The effect of magnetism on the atomic relaxations at the Co/Cu interface is revealed.