Within the aim of a dynamical study of on- and off-shell heavy quarks Q in the quark gluon plasma (QGP) - as produced in relativistic nucleus-nucleus collisions - we study the heavy quark collisional scattering on partons of the QGP. The elastic cross sections $\sigma_{q,g-Q}$ are evaluated for perturbative partons (massless on-shell particles) and for dynamical quasi-particles (massive off-shell particles as described by the dynamical quasi-particles model "DQPM") using the leading order Born diagrams. We demonstrate that the finite width of the quasi-particles in the DQPM has little influence on the cross sections $\sigma_{q,g-Q}$ except close to thresholds. We, furthermore, calculate the heavy quark relaxation time as a function of temperature T within the different approaches using these cross sections.