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 P-X(beta) of the quantity beta 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 P-X(beta) 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.