To demonstrate the potential of nanopores in bilayer graphene for DNA sequencing, we computed the current-voltage characteristics of a bilayer graphene junction containing a nanopore and found that they change significantly when nucleobases are transported through the pore. To demonstrate the sensitivity and selectivity of example devices, we computed the probability distribution PX(β) of the quantity β representing the change in the logarithmic current through the pore due to the presence of a nucleobase X (X = adenine, thymine, guanine, or cytosine). We quantified the selectivity of the bilayer-graphene nanopores by showing that PX(β) exhibits distinct peaks for each base X. To demonstrate that such discriminating sensing is a general feature of bilayer nanopores, the well-separated positions of these peaks were shown to be present for different pores, with alternative examples of electrical contacts.