Yes this is theoretically possible, and it is also observed in practice (in particular on ship propellers where it is called cavitation). Gases, including oxygen and carbon dioxide certainly, and probably also nitrogen, are absorbed in solution by water from the air (if oxygen were not absorbed fish would not be able to breathe underwater). Once in solution, the distribution of dissolved particles diffuses naturally through the whole body of the water, to any depth.
If water is agitated into turbulent motion, then gases are ejected from solution because turbulence generates rapidly varying pressure. Gas is less soluble in liquid at lower pressure. You can also observe the effect simply by shaking a fizzy drink.