Old World finch (6)
I believe the answer is:
'old world finch' is the definition.
(I've seen this in another clue)
This is the entire clue.
(Other definitions for linnet that I've seen before include "See three down" , "Finch formerly popular as a pet" , "Brown-gray kind of finch" , "Member of finch family" , "Songster" .)