The most common and most likely answer to this crossword clue is the 7 letter word NODDING
I've seen this crossword clue in The Daily Telegraph
N
O
D
D
I
N
G
Buddy explains!
The straight part
We know that "One’s falling asleep" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one’s falling asleep = nodding.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 NODDING mean?
NODDING verb- To incline the head up and down, as to indicate agreement.
- To briefly incline the head downwards as a cursory greeting.
- To sway, move up and down.
- To gradually fall asleep.
- To signify by a nod.
- To make a mistake by being temporarily inattentive or tired
- To head; to strike the ball with one's head.
- To allude to something.
- To fall asleep while under the influence of opiates.
- The act by which one nods.
- Having the flower facing downwards.