In this paper we discuss interplays between the Aharonov-Bohm effect and the transport properties in mesoscopic ring structures based on graphene. The interlayer interaction leads to a change of the electronic structure of bilayer graphene ring such that the electronic energy dispersion law exhibits a gap, either by doping one of the layers or by the application of an external perpendicular electric field. Gap adjustments can be done by varying the external electric field, which provides the possibility of obtaining mesoscopic devices based on the electronic properties of bilayer graphene. This opens the way to controllable manipulations of phase-coherent mesoscopic phenomena, as well as to Aharonov-Bohm oscillations depending on the height of the potential step and on the radius of the ring. For this purpose one resorts to a tight-binding model such as used to the description of conductance.