In this paper, we explored computationally the feasibility of modulating the bandgap in a single-wall BN nanotube (BNNT) upon noncovalent adsorption of organic molecules, combined with the application of a transverse electric field. Effects of analytes' physisorption on the surface of BNNTs regarding structural and electronic properties were delineated. Relatively large binding energies were calculated, however, with minimal perturbation of the structural framework. Electronic structure calculations indicated that the bandgap of BNNTs can be modified by weak adsorption due to the presence of adsorbate states in the gap of the host system. Furthermore, we have shown that the application of a transverse electric field can tune the bandgap by shifting adsorbate states, consistent with calculated current-voltage characteristics.