The left-back, who will finalise the switch when his contract expires at the end of June, has spent six years with Schalke and described the "many long sleepless nights" he endured before deciding to leave.
In a message posted on his Facebook page, the 23-year-old Bosnia-Herzegovina international thanked the Bundesliga club and their fans, while claiming he turned down "many financial offers from large European clubs" in the January transfer window.
"There has been a lot of discussion written and around lately," his post began. "My contract with my friends at Schalke expires at the end of the month and I have spent many long sleepless nights in the last few months with the considerations on how to continue.
"Because, honestly, you're amazing! As you have confessed to me in the last few years, it is indescribable! I thank you for that.
"In the last six years, starting from the Under-19s until today, I've become one of you. Anyone who once left his heart in the [club] will be a part of it and never forget this second home.
"As I said, I've been very torn in the last few months. But now I have decided and I owe it to you that you know it: I will continue my football career in London, at Arsenal. It hurts me to go away from Gelsenkirchen, but Arsenal and Arsene Wenger offer me the great chance to prove [myself].
"In the last winter, I have had many, many financial offers from large European clubs. I rejected everything then. I have always spoken with my club about it and finally [made a decision] as to how I should go on with my career.
"I fought [every] game to play for my club in order to achieve the maximum success [for] Schalke.
"Schalke 04 stays in my heart. I thank you for your encouragement, your love, the strength you gave me. I won't forget it all my life. I promise.
"And if I'm lucky enough to be on the road with the Gunners and come to Germany, I'm definitely going to pay you a visit."
Kolasinac is the first Arsenal signing since Wenger agreed a new two-year contract last month.