In an effort to produce Giant Magnetocaloric effect (GMCE) near room temperature, in a first ever such study, the austenite transformation temperature (A_s) was fine tuned to ferromagnetic Curie temperature (T_C) in Ferromagnetic Shape Memory Alloys (FSMA) and a large GMCE of ΔSM = -81.8 J/Kg-K was achieved in Ni₅₀Mn_18.5Cu_6.5Ga₂₅ alloy during reverse martensitic transformation (heating cycle) for a magnetic field change of 9 T at 303 K. Fine tuning of A_s with T_C was achieved by Cu substitution in Ni₅₀Mn_25-xCu_xGa₂₅ (0 ≤ x ≤ 7.0)-based FSMAs. Characterizations of these alloys were carried out using Optical and Scanning Electron Microscopy, X-ray Diffraction (XRD), Differential Scanning Calorimetry (DSC) and DC magnetization measurements. Addition of Cu to stoichiometric Heusler type Ni₂MnGa increases the martensitic transformation temperatures and decreases TC. Concurrently, DSM increases with Cu addition and peaks at 6.5 at% Cu for which there is a virtual overlap between T_C and A_s. Maximum Refrigerant Capacity (RCP) of 327.0 J/Kg was also achieved in the heating cycle for 9 T field change at 303 K. Corresponding values for the cooling cycle measurements (measured during forward transformation) were 30.4 J/Kg-K and 123.5 J/Kg respectively for the same 6.5 at% Cu sample under the same thermo-magnetic conditions.