If you put something about, you tell it to people that you meet and cause it to become well-known. (The forms put around and put round are also used in British English. mainly BRIT)
Moderates are putting it about that people shouldn’t take the things said at the Republican Convention too seriously...
The King had been putting about lurid rumours for months.
PHRASAL VERB: V it P that, V P n (not pron), also V n P