Despite Konami and Kojima announcing that Big Boss would be played by 24 actor, Kiefer Sutherland, it seems that David Hayter may indeed reprise his role as David, aka Solid Snake.
As noted on the official MGS5: The Phantom Pain IMDB page, David Hayter will be voicing a character named “David/Big Boss”. If you didn’t know this, David is Solid Snake’s original name before given his code name. Now of course anyone on IMDB can submit edits, but unlike Wikipedia they must be approved. So it’s possible that this could just be a listed mistake, though given the time line of the game it wouldn’t be surprising is Solid Snake made an appearance towards the end of the game.
Thanks youtube user, YongYea.