Optical fiber optrodes are attractive sensing devices due to their ability to perform point measurement in remote locations. Mostly, they are oriented to biochemical sensing, quite often supported by fluorescent and spectroscopic techniques, but with the refractometric approach considered as well when the objective is of high measurement performance, particularly when the focus is on enhancing the measurand resolution. In this work, we address this subject, proposing and analyzing the characteristics of a fiber optic optrode relying on plasmonic interaction. A linearly tapered optical fiber tip is covered by a double overlay: the inner one - a silver thin film and over it - a dielectric layer, with this combination allowing to achieve, at a specific wavelength range, surface plasmonic resonance (SPR) interaction sensitive to the refractive index of the surrounding medium. Typically, the interrogation of the SPR sensing structures is performed, considering spectroscopic techniques, but in principle, a far better performance can be obtained, considering the reading of the phase of the light at a specific wavelength located within the spectral plasmonic resonance. This is the approach which is studied here in the context of the proposed optical fiber optrode configuration. The analysis performed shows the combination of a silver inner layer with a dielectric titanium oxide layer with tuned thicknesses enables sensitive phase reading and allows the operation of the fiber optic optrode sensor in the third telecommunication wavelength window.