Guilherme Rambo of 9to5Mac has done some spelunking and has found evidence that Apple is currently planning to charge $4.99 for Apple Arcade, due later this year:
Today, I was able to get information about the price of an Apple Arcade subscription to customers. This information is available in one of the APIs used by the App Store app. According to a promotional message found in the service, the price for Apple Arcade will be $4.99 / month, including a one-month free trial. As Apple previously announced, the service will allow access to all members in a Family Sharing account.
I suppose not every Apple service can cost $9.99. I do wonder if Apple Arcade will be offered with other Apple services in a bundle, but $4.99 seems like a real bargain if Apple Arcade lives up to its promise.