Not baseless assumptions, there are hints in the manga to what he just , Itachi's proficiency in Body flicker and Genjutsu in comparison to Shishui's known mastery in Body flicker and Genjutsu. Itachi looking at him like an older brother.. It's not a baseless assumption to think they trained together... Considering how Itachi turned out... and he even learned his way of a ninja from directly copying shishui's..
the baseless assumptions is that Obito thought Itach izamai,, when It was stated that itachi was reading up on clan history etc from a little child... No where was the opposite stated...
Not getting into an argument for those who think Obito actually taught Itachi..
Hint hint, everything Itachi said there was a lie
Madara stealing his brothers eyes
Obito being his teacher
Obito attacking the village (that wasn't a lie)
But if you don't get the theme yet, Itachi was painting Madara in a picture of evil so Sasuke would either stay away or go after him..