First of all we would have to define what we consider sleeping. As humans, we associate sleeping with closing our eyelids, not moving our bodies, and, in many cases, lying down. When it’s dark out, our pineal gland releases melatonin and our body recognizes that it’s time to “shut down the neurons.” If we were to measure the brain’s electrical activity with an EEG when we’re asleep, we would identify a different pattern from the one we would see if we were awake.
Tuna don’t have eyelids, so their eyes are always open. That, in and of itself, doesn’t necessarily mean they’re not asleep: we can’t close our ears, for example, and that doesn’t stop us from sleeping. Tuna can’t ever stop swimming, because they need the oxygen they get from filtering water while they move. If you look into a tank with tuna larvae or juveniles at night, you see they move more slowly, but although we’ve tried, we haven’t yet managed to measure their swimming speed in the dark, nor have we been able to record them without any light. And it appears that the brain activity we associate with sleep in mammals is different from that of fish. There are indications that tuna release melatonin, but we’re not yet sure about the relationship between their daily rhythm and that secretion. So for now all we can say is that, although there’s no scientific evidence of their sleeping, they probably enter some similar state in which they rest and save energy.