Which grammar books say it can be omitted? I wouldn't. It's something you might hear, but I would not write the sentence without it.

English Teacher
"It was you that told me to say so"
The issue of omission of that in this sentence varies depending on grammar books.
I wonder what you teachers think.
Personally I don't think it can be omitted.
Which grammar books say it can be omitted? I wouldn't. It's something you might hear, but I would not write the sentence without it.
"It was you who told me to say that" is also possible. Whether you use "who" or "that", I can't see a way of omitting it.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.