The high quality contact between graphene and the metal electrode is a crucial factor in achieving the high performance of graphene transistors. However, there is not sufficient research about contact resistance reduction methods to improve the junction of metal-graphene. In this paper, we propose a new method to decrease the contact resistance between graphene and metal using directly grown graphene over a metal surface. The study found that the grown graphene over copper, as an intermediate layer between the copper and the transferred graphene, reduces contact resistance, and that the adhesion strength between graphene and metal becomes stronger. The results confirmed the contact resistance of the metal-graphene of the proposed structure is nearly half that of the conventional contact structure.