Unofficially, in confidence, not for publication, as in
What he was about to say, he told the reporters, was strictly off the record. Probably alluding to striking evidence from a court record (because it is irrelevant or improper), this term came into wide use in the mid-1900s, especially with reference to persons who did not wish to be quoted by journalists. For antonyms, see
go on record;
just for the record.