  We obtain two exact solutions of Einstein gravity coupled to nonlinear
electrodynamics (NLED) in $(2+ 1)$-dimensional Anti-de Sitter (AdS) spacetime.
The solutions are characterized by the mass $M$, angular momentum $J$,
cosmological constant or (anti) de Sitter parameter $\Lambda$, and an
electromagnetic parameter $Q$, that is related to an electric field in the
first solution and to a magnetic charge for the second solution. Depending on
the range of the parameters, the solutions admit a charged rotating
asymptotically AdS black hole (BH) interpretation or a charged rotating
asymptotically AdS traversable wormhole (WH). If the electromagnetic field is
turned off, the stationary Ba\~nados-Teitelboim-Zanelli (BTZ) BH is recovered;
in such a way that our BH-WH solutions are nonlinear charged generalizations of
the stationary BTZ-BH. Moreover, in contrast to the BTZ metric, the derived AdS
solutions are singular at certain radius $r_{s} \neq 0$, resembling the ring
singularity of the Kerr-Newman spacetime; while if $\Lambda$ is positive the
curvature invariants of the second solution are finite.
