  Gravastar models have recently been proposed as an alternative to black
holes, mainly to avoid the pro\-ble\-ma\-tic issues associated with event
horizons and singularities. In this work, a regular variety of gravastar models
within the context of Einstein-Born-Infeld (EBI) nonlinear electrodynamics are
builded. These models presented here are truly regular in the sense that both
the metric and its derivatives are continuous throughout space-time, contrary
to other cases in the literature where matching conditions are necessary in the
interior and exterior regions of the event horizon. We investigated the
accretion process for spherically symmetric space-time geometries generated for
a nonlinear electromagnetic field where the energy momentum tensor have the
same form that an anisotropic fluid that is the general EBI case. We analyse
this procedure using the most general static spherically symmetric metric
ansatz. In this theoretical context, we examined the accretion process for
specific sphe\-ri\-ca\-lly symmetric compact configuration obtaining the
accretion rates and the accretion velocities during the process and the flow of
the fluid around the black hole. In addition, we study the behaviour of the
rate of change of the mass for each chosen metric.
