J’ai sommeil.

This is one of those very common differences between English and French.

In English, you say I am sleepy.
In French, the usual expression is J’ai sommeil, literally I have sleepiness.

French often uses avoir where English uses to be:

• J’ai faim = I’m hungry
• J’ai soif = I’m thirsty
• J’ai chaud = I’m hot
• J’ai sommeil = I’m sleepy

So even though it feels strange from an English point of view, J’ai sommeil is the normal idiomatic way to say it.

Sommeil means sleep or sleepiness, depending on context.

In J’ai sommeil, it does not mean that you are literally sleeping. It means you feel sleepy or drowsy.

So:

• J’ai sommeil = I’m sleepy / I feel sleepy

It describes a feeling, not the action of sleeping.

French does not like putting certain vowel sounds next to each other.
When je comes before a verb beginning with a vowel, the e drops and is replaced by an apostrophe.

So:

• je + ai becomes j’ai

This is called elision.

You will see the same thing in many other places:

• j’aime
• j’habite
• j’arrive

So J’ai sommeil is just the normal contracted form of Je ai sommeil, and Je ai is not used.

In this expression, French simply says avoir sommeil with no article.

So the fixed expression is:

• avoir sommeil = to be sleepy

If you add an article, the meaning changes:

• J’ai le sommeil léger = I’m a light sleeper
• J’ai un sommeil agité = I have restless sleep

But for the simple idea I’m sleepy, you just say:

• J’ai sommeil

A simple English-friendly approximation is:

• J’ai ≈ zhay
• sommeil ≈ so-may

So the whole sentence sounds roughly like:

• zhay so-may

A few pronunciation notes:

• The j in j’ai sounds like the s in measure
• The final l in sommeil is not pronounced the way English speakers might expect
• The eil ending sounds roughly like ay

A more accurate IPA pronunciation is:

• /ʒe sɔ.mɛj/

Yes, very natural.

It is a standard, everyday way to say I’m sleepy.

Depending on the situation, French speakers might also say things like:

• Je suis fatigué(e) = I’m tired
• Je suis crevé(e) = I’m exhausted or I’m wiped out
• Je meurs de sommeil = I’m dying of sleepiness / I’m so sleepy

But J’ai sommeil is completely normal and common.

They are related, but not identical.

• J’ai sommeil = I’m sleepy
  You feel like sleeping.
• Je suis fatigué(e) = I’m tired
  You feel tired in general, not necessarily ready to fall asleep.

For example:

• After lunch, you might say J’ai sommeil
• After a long day of work, you might say Je suis fatigué(e)

Sometimes both can be true, but they are not exact synonyms.

You can, but it is less common in everyday conversation.

• Je suis somnolent(e) means I am drowsy / sleepy
• J’ai sommeil is the more usual everyday expression

So if you want the most natural conversational French, J’ai sommeil is the safest choice.

Also remember that somnolent changes form:

• somnolent for a man
• somnolente for a woman

But J’ai sommeil stays the same for everyone.

No.

J’ai sommeil means I feel sleepy.
It does not mean that you are currently asleep.

If you want to say I’m sleeping, you would use the verb dormir:

• Je dors = I’m sleeping / I sleep

So:

• J’ai sommeil = I feel like sleeping
• Je dors = I am actually sleeping

You can do both very easily.

Negative:

• Je n’ai pas sommeil = I’m not sleepy

Question:

• Tu as sommeil ? = Are you sleepy?
• Est-ce que tu as sommeil ? = Are you sleepy?
• As-tu sommeil ? = a more formal/inverted version

So the expression works just like other avoir expressions.

You can add adverbs or expressions of degree.

Common examples:

• J’ai un peu sommeil = I’m a little sleepy
• J’ai vraiment sommeil = I’m really sleepy
• J’ai très sommeil = sometimes heard, but less standard
• J’ai tellement sommeil = I’m so sleepy
• J’ai trop sommeil = I’m too sleepy / so sleepy in casual speech
• Je meurs de sommeil = I’m dying of sleepiness

For a learner, very natural choices are:

• J’ai un peu sommeil
• J’ai vraiment sommeil
• J’ai tellement sommeil

Yes. It is best learned as a set expression.

Instead of translating word by word from English, memorize:

• avoir sommeil = to be sleepy

That will help you avoid the common mistake of trying to say something like je suis sommeil, which is not correct.

It is similar to memorizing:

• avoir faim
• avoir soif
• avoir peur

French uses many such expressions, and learning them as whole chunks is very helpful.