We propose a compact interferometer based on a dual-core transversally chirped microstructured optical fiber (MOF), which can be used as a highly sensitive, label-free, and selective biosensor. The air holes of the rings surrounding one core of the structure are functionalized for antibody detection by immobilization of an antigen sensor layer onto the walls of the holes. Changes in the thickness of the biomolecule layer can be detected as a function of the resonant wavelength in the transmittance spectrum. Numerical results demonstrate that the proposed biosensor could achieve a sensitivity of up to 5.6 nm/nm at 1600 nm and 4.7 nm/nm at 1440 nm, respectively. The sensor operation was optimized for the IR region, which allows the detection of biological agents.