In this work, we designed structures based on copper nanosubstrate with graphene and two-dimensional transition metal dichalcogenides (TMDC) in order to achieve an ultrasensitive surface plasmon resonance biosensor. This system contains seven components: SF11 triangular prism, BK-7 glass, Chromium (Cr) adhesion layer, thin copper film, layers of one of the types of transition metal dichalcogenides: MoS2, MoSe2, WS2 or WSe2 (defined as MX2), graphene, sensing layer with biomolecular analyte. Copper was chosen as a plasmonic material because it has a higher conductivity than gold which is commonly used in plasmonic sensors. Moreover, copper is a cheap and widespread material that is easy to produce on a large scale. We have carried out both theoretical and numerical sensitivity calculations of these kinds of structures using the Goos–Hänchen (GH) shift method. GH shift is lateral position displacement of the p-polarized reflected beam from a boundary of two media having different indices of refraction under total internal reflection condition and its value can be retrieved from the phase change of the beam. The SPR signal based on the GH shift is much more sensitive compared to other methods, including angular and wavelength scanning, due to much more abrupt phase change of the SPR reflected light than its intensity ones. By optimizing the parameters of the SPR sensing substrate, such as thickness of copper, number of layers of 2D materials and excitation wavelength, we theoretically showed an enhanced sensitivity with a detection limit 10−9 refractive index unit (RIU).