A criminal denies offering 'spice' ... (7)
I believe the answer is:
aniseed
'spice' is the definition.
(I know that aniseed is a spice)
'a criminal denies' is the wordplay.
'criminal' is an anagram indicator.
'denies' with letters rearranged gives 'niseed'.
'a'+'niseed'='ANISEED'
'offering' acts as a link.
(Other definitions for aniseed that I've seen before include "Aromatic flavouring" , "Flavouring in pastis" , "Flavouring herb" , "Liquorice-flavoured seeds" , "Flavouring of sambuca" .)