Asymptotic Reissner–Nordström black holes

•We investigate two types of the BI-like nonlinear electromagnetic fields in the Einsteinian gravity.•We analyze the effects of nonlinearity on the electromagnetic field.•We examine the influences of the nonlinearity on the geometric properties of the black hole solutions.

We consider two types of Born–Infeld like nonlinear electromagnetic fields and obtain their interesting black hole solutions. The asymptotic behavior of these solutions is the same as that of a Reissner–Nordström black hole. We investigate the geometric properties of the solutions and find that depending on the value of the nonlinearity parameter, the singularity covered with various horizons.