On Azure, I am currently using my \"3-month Free Trial\" subscription, and just recently I got access to a \"Windows Azure MSDN - Visual Studio Ultimate\" subscription.
I don't know a way to do this from the portal but I do recommend using Windows Azure PowerShell which has the ability to manage multiple subscriptions and in your case you can just do this cmdlet to copy a service into another subscription:
Publish-AzureServiceProject -sn MyNewSubscriptionName
There is no way you can accomplish in the Portal interface. You do have an option to request Windows Azure Billing Team and request them move all of your currently configured and running services from one Subscription to other subscription, the key is "all".
IF you want to move one specific service from one subscription to another then the available option is to deploy directly to other subscription.
I am battling with azure support right now on trying to do the same thing. From what I can tell your best bet is creating the services on your MSDN subscription and redeploying.
I did do this once. I had to contact their billing team and after a couple of days of back and forth between them it was taking too long so I just did it manually, redeploying the sites to the new subscription. This may not be possible if you have many sites. I did a backup and restore of the SQL Azure database.