If I am not mistaken, "asleep" is a predicative adjective (never an adverb) which is usually used as a part of a predicate in a sentence (after a verb). He is asleep. He has fallen asleep. Sometimes this adjective can be placed after a noun: I saw a man asleep.

I have just read an article about the word "asleep". It says that there's no agreement among linguists whether "asleep" is only an adjective or can be both an adjective and an adverb. Dictionaries also define this word differently, for example: asleep - The Oxford English Dictionary - Adv., pr. Adj.; Webster's New Collegiate Dictionary - Adj. The definition of "asleep" as an adjective is the following: 1: being in a state of sleep 2: dead 3: lacking sensation: numb 4 a: inactive, dormant b: not alert: indifferent And here's the definition of it as an adverb: 1: into a state of sleep 2: into the sleep of death 3: into a state of inactivity, sluggishness, or indifference Still, the point of view that "asleep" can be an adverb is not common.

Adverb is used to describe how a certain action (verb) is done. He fell (verb) asleep (adverb) quickly. Their anxieties were put (verb) asleep (adverb). The two examples are taken from my dictionary.

Your example is the adverb.

Hi, "He fell asleep." Source: My Dictionary