The linear perturbations evolving on streamwise boundary layer streaks which yield maximum energy growth are computed. The steady and spanwise-periodic streaks arising from the nonlinear saturation of optimally growing streamwise vortices are considered as base flow. It is shown that significant transient growth may occur for both sinuous antisymmetric perturbations and for varicose symmetric modes. The energy growth is observed at amplitudes significantly below the threshold beyond which the streaks become linearly unstable and is largest for sinuous perturbations, to which the base flow considered first become unstable. The optimal initial condition consists of velocity perturbations localized in the regions of highest shear of the streak base flow, tilted upstream from the wall. The optimal response is still localized in the areas of largest shear but it is tilted in the flow direction. The most amplified perturbations closely resemble the unstable eigenfunctions obtained for streaks of higher amplitudes. The results suggest the possibility of a transition scenario characterized by the non-modal growth of primary perturbations, the streaks, followed by the secondary transient growth of higher frequency perturbations. The implication for turbulent flow is also discussed.