Different-temperature Cu/NbTi cladding extrusion was put forward where Cu and NbTi are different in temperature during plastic deformation. Different-temperature Cu/NbTi cladding extrusion can significantly lower the deformation temperature of Cu cladding layer so as to reduce the difference in yield stresses between Cu cladding layer and NbTi alloy core, which contributes to accommodating the deformation of the two metals. Rigid viscoplastic finite element method was used to simulate different-temperature Cu/NbTi cladding extrusion in order to reveal the interface compatibility between Cu cladding layer and NbTi alloy core, where various cone angles of bottom die, including 60°, 120° and 180°, were adopted. The results show that increasing the cone angle of bottom die contributes to reducing the relative elongation between Cu cladding layer and NbTi alloy core, which is conducive to the interface bonding between Cu cladding layer and NbTi alloy core. According to the optimal parameters from finite element simulation, Cu/NbTi cladding extrusion die with the cone angle of 180° can be used to implement Cu/NbTi cladding extrusion experiment. The experimental results show that stable flow of metal takes place during Cu/NbTi cladding extrusion, where Cu cladding layer and NbTi alloy core present compatible deformation and the relative elongation between them is small. The experimental results agree well with the simulated ones.