The electron-phonon interaction at the interface between topological insulator (TI), namely, Bi2Se3 and Bi2Te3 two-dimensional (2D) nanoflakes, to a gold substrate as a function of TI flake thickness is studied by means of Raman scattering. We reveal the presence of interface-enhanced Raman scattering and a strong phonon renormalization induced by carriers injected from the gold substrate to the topological surface in contact. We derive the change of the electron-phonon coupling showing a nearly linear behavior as a function of layer thickness. The strongly nonlinear change of the Raman scattering cross section as a function of flake thickness can be associated with band bending effects at the metal-TI interface. Our results provide spectroscopic evidence for a strongly modified band structure in the first few quintuple layers of Bi2Se3 and Bi2Te3 in contact with gold.