Drop off nearly twelve (4)
I believe the answer is:
doze
'drop off' is the definition.
(thesaurus)
'nearly twelve' is the wordplay.
'nearly' means to remove the last letter.
'twelve' becomes 'dozen' ('dozen' can be a synonym of 'twelve').
'dozen' with its final letter removed is 'DOZE'.
(Other definitions for doze that I've seen before include "little sleep" , "Catnap" , "A short light sleep" , "Go to sleep" , "Sleep fitfully" .)