Offensive rumour fuelled by opprobrium in the extreme (7)
I believe the answer is:
noisome
'offensive' is the definition.
(I know that offensive can be written as noisome)
'rumour fuelled by opprobrium in the extreme' is the wordplay.
'rumour' becomes 'noise' (I've seen this in other clues).
'fuelled by' means one lot of letters goes inside another (inserted letters go into or 'fuel' others).
'in the extreme' says to hollow out the word (remove centre letters) (the letters on the extremes of the word are used).
'opprobrium' with its centre taken out is 'om'.
'noise' going around 'om' is 'NOISOME'.
(Other definitions for noisome that I've seen before include "Foul; putrid" , "Smelling bad" , "Disgusting to sight or smell" , "Foul-smelling - semi-Ono (anag)" , "Odious" .)