Growth of REBa2Cu3Oy (REBCO, RE=Y, Nd) crystals on the MgO substrates by the liquid phase epitaxy (LPE) process was investigated in order to clarify the growth mechanism. The in-plane alignment of crystal orientation was improved during the LPE process due to preferential dissolution and growth, even from the polycrystalline seed film. It was found that the orientation of preferential growth depended on the kind of RE for the REBCO system. The phenomenon could be explained by the coarsening mechanism, considering the interfacial energies between the REBCO and the MgO crystals. The preferential growth mechanism could be understood by considering both the lattice matching and the Coulomb force between the REBCO crystal and the substrate materials at the interface. The calculation results were in good agreement with the experimental results.