Cornell potential is known to represent the quark-antiquark confinement interaction. In addition to the Cornell potential, there have been other interactions in the literature that demonstrate confining structure in the quark-antiquark system. Guendelman has proposed a nonlinear electrodynamics (NED) model which results in electric potential in the form of Cornell interaction. In this study, we propose two NED models whose electric potential is in the form of two other confining potentials. We also study the coupling of gravity to one of these models which are in the form of a correction to linear Maxwell's theory. We find spacetime which can be a black hole or a naked singular particle. In the zeroth approximation, we examine the circular speed of a distant test particle orbiting the central black hole. The Newtonian potential due to the spacetime at the location of the distant particle possesses an additional term proportional to the NED parameter. Due to the presence of such a term, the speed-radius curve gets improved from the pure Reissner-Nordstrom. It is a kind of weaker confinement acting on distant parts of a galaxy due to a supermassive central black hole.