Characters in the TV show pronounce it ass-shy or ah-shy.

In audiobooks, Asshai is pronounced as ASH-eye or a-SHY, depending on the narrator's accent.

In the HBO's show Game of Thrones, it is pronounced as uh-SHY.

For your reference, you may check these pronunciation guides:

• A Wiki of Ice and Fire (Pronunciation Guide)

• Official Pronunciation Guide of Game of Thrones

Consult with the following table:

You can refer to this answer to see how can you convert IPA to voice. The table was generated as per AWOIAF Pronunciation guide. I will add GRRM's version once I can find a video of him saying that word.

However you like

George R. R. Martin has never been one for pronunciations and has said a few times that you can pronounce anything however you like.

Favorite character?

GRRM : Tyrion in Ice and Fire. Abner Marsh in Fevre Dream.

Glad to hear you pronounce the names

GRRM: In my youth I had a strong NJ accent, only reader in family, knew a lot of words that I had never heard spoken aloud. When I went away to college I found I was pronouncing a lot of these words wrong. I came to not care much about pronunciation. Pronounce the names of my characters however you like.

There are dangers in being a gardener, the story can run away from you - Shakespeare had to kill Mercutio because he was taking over play.

_{So Spake Martin, ODYSSEY CON 2008 (MADISON, WI; APRIL 4-6 2008)}