Aspirant is quiet about offer (9)
I believe the answer is:
pretender
'aspirant' is the definition.
(thesaurus)
'quiet about offer' is the wordplay.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'about' becomes 're' ('re' means regarding or about).
'offer' becomes 'tender' (tender can mean to offer something).
'p'+'re'+'tender'='PRETENDER'
'is' is the link.
(Other definitions for pretender that I've seen before include "One claiming a title" , "One who claims a title or position" , "fake" , "Claimant to a throne, usually without just title" , "Claimant, like Bonnie Prince Charlie" .)