The (attractive) magnetic force on two parallel wires can be seen by the two wires bending towards each other. This can be explained by the Lorentz force acting on the charges and the charges "pulling" transversally on the wires, because they cannot leave the wire because of the work needed to leave the wire.
Now the charges inside of a wire connected to the ends of a voltage source also experience a (generally much stronger) electric force along the main axis of the wire. As they collide with the atoms of the wire, shouldn't they "push" the wire in the direction of the positive cathode? Can this strain be measured?