This work aims at examining the thermophoresis and Brownian motion characteristics on the stagnation point flow of Cross nanofluid over a permeable stretching /contracting surface. Flow equations are modelled by means of the Buongiorno nanofluid model. The transformed non-linear fluid transport equations are solved numerically through Runge-Kutta Fehlberg (RKF) approach. The temporal stability test is executed to reveal the behaviour of dual solution that arises for various initial guesses. To validate the present model, first and second solutions are compared with earlier published works which found good agreement. The fluctuations of velocity, heat and mass transfer distributions are scrutinized through the graphs with active parameters such as Wessienburg number, Brownian motion, Eckert number, and thermophoresis. The results exposed that the Brownian motion declines the rate of heat transfer for the first solution. Cross nanofluid velocity amplifies by rising Weissenberg number. By the impact of Eckert number, the second solution has a higher magnitude than the first solution. For the first solution, the cross nanofluid velocity rises when increasing the suction/injection parameter.