This manuscript presents the development of a low-pressure extrinsic Fabry-Perot fiber optic sensor based on a thin polyester film, using a phase signal analysis. The proposed interferometer is controlled by the simple contact interaction between the polymer membrane and a multimode fiber optic tip. The created cavity was uniformly stressed by applying a pressure varying from 0 to 2 psi. A finite element analysis was performed for these parameters. The stress applied to the membrane was below the yielding point. Thus, a linear study could be performed. The wavelength spectra exhibited a blue shift with a sensitivity of around 10.5 nm/psi. Although the interference spectra presented crosstalk measurements and required an initial set-point calibration, these effects were overcome by examining the spatial frequency components’ phase analysis. The sensors offered a high sensitivity, close to 3.5 rad/psi. Ultimately, this sensor is a high sensitivity and versatile alternative for low-pressure detection