Lions and bears have come into contact by way of human captivity, and news reports of fights show that either animal is capable of winning in an altercation. The results of the fights are mixed and seem to vary greatly depending on the subspecies of lion and bear involved.
There are a handful of records of fights between lions and bears from ancient Rome and from the late 1800s in California and Mexico. There is also at least one record of an unintended fight between a lion and a bear breaking out at a circus, according to History Banter. These records indicate mixed results and are not entirely reliable. Since lions and bears do not occupy the same territory in the modern world, it would take questionable acts of animal cruelty to conclusively answer this question.