Neutrino oscillations, a fundamental discovery in particle physics, suggest that neutrinos have non-zero masses. This opens the possibility of physics beyond the Standard Model, including pseudo-Dirac neutrinos, which oscillate between active and sterile states over cosmological distances. In this study, we analyze 10 years of high-energy astrophysical neutrino data from the IceCube Neutrino Observatory, focusing on prominent sources like NGC 1068, TXS 0506+056, and PKS 1424+240, to search for signatures of pseudo-Dirac neutrino oscillations. By examining event distributions, we constrain the mass-squared differences ($\delta m^2$) between active and sterile states. We find no significant deviations from the Standard Model, placing a 90\% confidence level limit of $\delta m^2 < 4\times 10^{-19}~eV^2$ for NGC 1068. These results offer new insights into the potential existence of pseudo-Dirac neutrinos and demonstrate the unique ability of IceCube to probe neutrino physics over vast cosmic distances. Future analyses with larger datasets could improve sensitivity to even smaller $\delta m^2$ values.