We study the superlattice minibands produced by the interplay between a moire pattern induced by a hexagonal BN substrate on a graphene layer and the interlayer coupling in bilayer graphene with Bernal stacking (BLG). We compare moire miniband features and zero-energy gaps in BLG, where they are affected by the interlayer asymmetry of BLG/h-BN heterostructure and trigonal warping characteristic for electrons in Bernal-stacked bilayers, with those found in monolayer graphene.