Eager to have paintings around study (6)
I believe the answer is:
ardent
'eager' is the definition.
(synonyms)
'paintings around study' is the wordplay.
'paintings' becomes 'art' (painting is an art).
'around' is an insertion indicator.
'study' becomes 'den' (term for a study or office).
'art' going around 'den' is 'ARDENT'.
'to have' is the link.
(Other definitions for ardent that I've seen before include "Ranted, being passionate" , "Passionate and glowing" , "Fiery and passionate" , "Intensely enthusiastic" , "Ranted, being impassioned" .)