It is debated whether different forms of bistable perception result from common or separate neural mechanisms. Binocular rivalry involves perceptual alternations between competing monocular images, whereas ambiguous figures such as the Necker cube lead to alternations between two possible pictorial interpretations. Previous studies have shown that observers can voluntarily control the alternation rate of both rivalry and Necker cube reversal, perhaps suggesting that bistable perception results from a common mechanism of top-down selection. However, according to the biased competition model of selective attention, attention should be able to enhance the attended percept and suppress the unattended percept. Here, we investigated selective attentional modulation of dominance durations in bistable perception. Observers consistently showed much weaker selective attentional control for rivalry than for Necker cube reversal, even for rivalry displays that maximized the opportunities for feature-, object-, or space-based attentional selection. In contrast, nonselective control of alternation rate was comparably strong for both forms of bistable perception and corresponded poorly with estimates of selective attentional control. Our results support the notion that binocular rivalry involves a more automatic, stimulus-driven form of visual competition than Necker cube reversal, and as a consequence, is less easily biased by selective attention.