The most common and most likely answer to this crossword clue is the 10 letter word SLUMBERING
I've seen this crossword clue in The Daily Telegraph
S
L
U
M
B
E
R
I
N
G
More clues leading to the result SLUMBERING
Buddy explains!

The straight part
We know that "Asleep" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case asleep = slumbering.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 SLUMBERING mean?
SLUMBERING verb- To be in a very light state of sleep, almost awake.
- To be inactive or negligent.
- To lay to sleep.
- To stun; to stupefy.
- Sleep
- Temporarily inactive.