OFFHAND, OFF HEAD, BY HEART

English for Today by Ganiu Abisoye Bamgbose (Dr GAB)

OFFHAND, OFF HEAD, BY HEART

Offhand means to be unfriendly, and to show little interest in other people in a way that seems slightly rude:
She isn’t usually offhand; perhaps she wasn’t just in a good mood.
Also, if you say something offhand, you say it without checking the details or facts of it: I couldn’t give the figures offhand (it means I can’t go ahead to give the figure without checking the record).

By heart: If you know something such as a poem by heart, you have learned it so well that you can remember it without having to read it:
I know the whole passage offhand. ✖️
I know the whole passage off head. ✖️
I know the whole passage by heart. ✔️

Off head is not a lexical item in English but If you say that someone is off their head, you think that their ideas or behaviour are very strange, foolish, or dangerous: He’s gone completely off his head.
GAB