Current nanotechnology provides the possibility of creating hybrid nanostructures composed of semiconductor quantum dots and metal nanoparticles. Due to the interaction of excitons with surface plasmons, several optical properties are strongly modified, attracting the scientific attention in quantum dot – metal nanoparticle hybrids, both theoretically and experimentally. A system that has attracted important attention in the literature is the asymmetric, tunneling-controlled, double semiconductor quantum dot, demonstrating interesting optical properties, such as tunneling-induced transparency, slow light, and Autler-Townes splitting, while the pump-probe optical response and the four-wave mixing in this nanostructure gives interesting results. In this study, we theoretically examine the pump-probe optical response in a complex structure composed of an asymmetric double semiconductor quantum dot molecule, which is coupled to a metal nanoparticle via a long-range Coulomb interaction. We aim at the investigation of the modified nonlinear pump-probe optical properties of a double semiconductor quantum dot molecule, due to the excitonic-plasmonic interaction. Starting with the Hamiltonian, in the dipole and the rotating wave approximations, we first derive the density-matrix equations treating the problem in the quasi-static limit, which are solved numerically and obtain the steady state response of the system. Next, we use the results from the density matrix elements to calculate the first-order susceptibility and present the absorption/dispersion spectra for the semiconductor quantum dot and the metal nanoparticle separately, as well as, for the whole hybrid system, as a function of the pump-probe field detuning. The dependence of the spectral response is investigated, while the interparticle distance varies, for different values of the physical quantities introduced in the problem, including the electron-tunnelling coupling rate, the energy difference between the upper states of the semiconductor quantum dot molecule and the pump-field detuning.