A look that inwardly may be a greeting (5)
I believe the answer is:
aloha
'a greeting' is the definition.
(I've seen this before)
'a look that inwardly' is the wordplay.
'look' becomes 'lo'.
'inwardly' indicates the central letters (only the letters inside the word are used).
The centre of 'that' is 'ha'.
'a'+'lo'+'ha'='ALOHA'
'may be' is the link.
(Other definitions for aloha that I've seen before include "Hawaiian hello or goodbye" , "A greeting in Hawaii" , "Welcome to Honolulu!" , "Hawaiian farewell" , "Hawaiian greeting" .)