[en] Within the 'pseudo-Dirac' scenario for massive neutrinos the existence of sterile neutrinos which are almost degenerate in mass with the active ones is hypothesized. The presence of these sterile neutrinos can affect the flavor composition of cosmic neutrinos arriving at Earth after traveling large distances from astrophysical objects. We examine the prospects of neutrino telescopes such as IceCube to probe the very tiny mass squared differences 10^-12 eV²>Δm²>10^-19 eV², by analyzing the ratio of μ-track events to showerlike events. Considering various sources of uncertainties which enter this analysis, we examine the capability of neutrino telescopes to verify the validity of the pseudo-Dirac neutrino scenario and especially to discriminate it from the conventional scenario with no sterile neutrino. We also discuss the robustness of our results with respect to the uncertainties in the initial flavor ratio of neutrinos at the source.