Flow of electrons in wire

If we apply potential difference through wire we know that there is an electric current which is the flow of electrons.I want to know how do electrons move do they move within wire like in the picture below or do they move only through its surface? . If it moves only through its surface why is there n which is number of electrons per cubic meter in I=qnav. All the pictures i have seen on the internet shows it in a way that electrons move in wire too. But some people I conversed with told me it only moves through its surface because there is no net electric field inside conductor.(isn't this applied on electrostatics when electrons don't move)