Lorentz force $\vec F$ for an external magnetic field $\vec B$ is $$ \vec F = q \vec v \times \vec B $$ and the magnitude of this force is $ F = (qv \times B) \sin \theta $.
According to this equations Lorentz force is perpendicular to both the magnetic field and the direction of the movement of the electrons. So the only possible way for a current carrying wire to exert a force on elements of the same wire is to bend one portion of the wire to some angle $ \theta$ with the other wire. Am I right? Is there any other way too?