To demonstrate the potential of nanopores in bilayer graphene for DNA sequencing, we compute the current-voltage characteristics of a bilayer graphene junction containing a nanopore and show that this changes significantly when nucleobases are transported through the pore. To demonstrate the sensitivity and selectivity of exemplar devices, we compute 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 (= adenine, thymine, guanine or cytosine). We quantify the selectivity of the bilayer-graphene nanopores by showing that Px(β) possesses 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 are shown to be present for different pores, with alternative examples of electrical contacts.