The most common and most likely answer to this crossword clue is the 6 letter word DROWSE
I've seen this crossword clue in The Daily Telegraph
D
R
O
W
S
E
More clues leading to the result DROWSE
Buddy explains!

The straight part
We know that "Nod off" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case nod off = drowse.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does DROWSE mean?
DROWSE noun- The state of being sleepy and inactive.
- To be sleepy and inactive.
- To nod off; to fall asleep.
- To advance drowsily. (Used especially in the phrase "drowse one's way" ⇒ sleepily make one's way.)
- To make heavy with sleepiness or imperfect sleep; to make dull or stupid.