Inside nanobubbles would probably be "a high-pressure gas". Whether it be an inert gas or vaporized surrounding liquid depends on the temperature, pressure, components, and many other experimental variables.
Regarding theory on nanobubbles, and starting from a classical, macro-scale standpoint, the Laplace pressure describes the pressure inside a bubble due to surface tension on the curved vapor-liquid surface. The pressure will be necessarily higher inside the bubble in order that it doesn't collapse. The back-of-the-envelope calculation on that Wikipedia page indicates around 10 bar for a 300 nm bubble in water.
For smaller nanobubbles, things get more interesting. The surface tension is not constant with curvature and at the extreme curvatures necessary for nanobubbles this will change things. A first-order approximation to surface tension corrections is the Tolman length. Also, at the higher range of pressures involved gases would not be ideal, so some equation of state is likely necessary to describe the interior phase. As things get even smaller, then you get into the details where the continuum approximations like surface tension break down entirely.
Song et al. comes to my mind of a paper where they look at nanobubbles in an actual experiment. I'm sure there are many other experiments in this area.