Use the Link to Game Center transfer option.
Source Device:
1. Sign in to the Game Center account you want to link with your game account.
2. Go to Menu > Account Transfer > Link to Game Center.
3. Check the player name and rank displayed on the account you want to transfer and tap "Link."
Once you have successfully linked your account, the word "Linked" will be displayed and the "Link to Game Center" button will change to "Unlink from Game Center".
*The option to connect to Game Center may not appear depending on your app settings.
Target Device:
1. Sign in to the Game Center account you have linked with your game on the source device.
2. Go to Menu > Account Transfer > Link to Game Center.
3. Check the player name and rank displayed on the account you want to import and tap "Account Transfer."