The bilayer and thin films of Bi(111) have demonstrated novel topological properties. Here, we investigate the electronic structures of Bi/Bi₂Te₃(111) and Bi/Al₂O₃(0001) by combining first-principles and tight-binding approximation calculations. Our results show that the Bi(111) bilayer is a semiconductor with a gap of about 0.2 eV. Its electronic states are strongly disturbed by the interaction with Bi₂Te₃(111) thin films, no matter whether the substrate has a band gap or Dirac surface state. Moreover, it is hard to see Rashba-type band splittings in such systems. In contrast, it demonstrates clean and giant Rashba-type splittings as strongly hybridized with insulating Al₂O₃(0001), which is due to the broken inversion symmetry induced by interfacing and the strong atomic spin-orbit coupling in Bi. Our tight-binding approximation analyses further reveal that the effect of substrate Al₂O₃(0001) on the band structure of the Bi(111) bilayer is equivalent to the action of external electric field in a range between 0.5 and 0.6 V/Å. Moreover, we find that the strong hybridization between Bi(111) bilayer and the electronic state of the substrate Bi₂Te₃(111) can lead to a topological phase transition, i.e. the change from a two-dimensional topological insulator into a mediocre insulator. Our study thus provides an insight into the interface-engineering of electronic states of Bi(111) bilayer.