The stagnation point flow of a nanofluid towards a permeable stretching/shrinking sheet using the Buongiorno's model is studied. Numerical results are obtained using boundary value problem solver bvp4c in MATLAB for several values of the governing parameters. The numerical results show that dual (upper and lower branch) solutions exist for the shrinking case, while for the stretching case, the solution is unique. A stability analysis is performed to determine the physical realizable in practice of the dual solutions. It is found that the skin friction decreases when the sheet is stretched, but increases when the suction effect is increased. It is also found that increasing the thermophoresis parameter reduces the heat transfer rate at the surface, while increasing the Brownian motion parameter increases the mass transfer rate at the surface.