Offer around right time (4)
I believe the answer is:
bird
'time' is the definition.
(both are slang for a prison sentence)
'offer around right' is the wordplay.
'offer' becomes 'bid' ('bid' can be a synonym of 'offer').
'around' indicates putting letters inside.
'right' becomes 'r' (common abbreviation).
'bid' enclosing 'r' is 'BIRD'.
(Other definitions for bird that I've seen before include "Prison time (slang)" , "Fowl" , "Avian" , "Egg-laying vertebrate" , "porridge" .)