Eagle’s attribute seen around river and island nest (5)
I believe the answer is:
eyrie
'nest' is the definition.
(eyrie is a kind of nest)
'eagle's attribute seen around river and island' is the wordplay.
'eagle's attribute' becomes 'eye'.
'seen around' means one lot of letters goes inside another.
'river' becomes 'r'.
'and' says to put letters next to each other.
'island' becomes 'i'.
'r'+'i'='ri'
'eye' going around 'ri' is 'EYRIE'.
(Other definitions for eyrie that I've seen before include "lofty house" , "Eagle's lair" , "animal home" , "place for the high and mighty" , "An eagle's nest (5)" .)