The most common and most likely answer to this crossword clue is the 2 plus 8 letter word IN EXTREMIS
I've seen this crossword clue in The Guardian
I
N
E
X
T
R
E
M
I
S
More clues leading to the result IN EXTREMIS
- One following US rock band is at death’s door
- One, following destruction of Reims, is close to death
- Beset by trouble, one’s bordering on careless almost
- Miners exit angrily, barely surviving
- One following on motorway is nearly late
- Anger about High St store, one stuck in M&S feeling desperation
- One nearest a certain phase of sleep is at the point of death
- At the last gasp I then may come over almost negligent
- One following rock band is near death
- One is about following eye movements at the last gasp
Buddy explains!

The straight part
We know that "close to death" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case close to death = in extremis.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.