Approaching eccentric grannie (7)
I believe the answer is:
nearing
'approaching' is the definition.
('near' can be a synonym of 'approach')
'eccentric grannie' is the wordplay.
'eccentric' indicates an anagram.
'grannie' is an anagram of 'NEARING'.
(Other definitions for nearing that I've seen before include "Moving towards" , "Approaching" , "Getting close to" , "Drawing closer to" .)