Due to their high intensity of emission in the O VI λλ1031.9 and 1037.6 lines, even small sunspots on the solar disk can strongly influence the intensity of the radiative scattering component of O VI lines in the corona. Observations of O VI disk spectra show a λ1032/λ1038 line intensity ratio of >2.6 in a sunspot, compared to quiet-disk values of ~2. The enhancement of the λ1032 line in comparison to the λ1038 line is likely due to interaction between molecular hydrogen emission from the sunspot and the chromospheric O5+. Modeling shows that a contribution from sunspots increases the coronal O VI λ1032/λ1038 intensity ratio to values considerably higher than those achieved with a quiet-disk or coronal hole spectrum. Therefore a reexamination of flow velocities derived from SOHO UVCS streamer observations must be made. This modeling demonstrates that the inclusion of sunspots, when present, may lead to nonzero outflow velocities at lower heights in streamer cores, in contrast to some existing model results.