Before you do it
You will have to make sure that both Apache web servers are the same version, I didn’t and it caused some problems.

My hosting company also mentioned that both cPanels must be the same version too, but I’m not sure about this.

You will also need root access to both servers and an SSH client like putty, so go ahead and connect to both servers.

Manually Moving WHM/cPanel accounts
Follow these steps:
1 – In your old server do:
/scripts/pkgacct “username” <------- Don't type the quotes

2 - After the first step is done do:
scp /home/username.tar.gz root@IP:/home/ <------- Where IP is the IP of your destination server

You will be asked for the root password.

3 - In your destination server do:
cd /home
/scripts/restorepkg “username” <------- Don't type the quotes

Disclaimer
Do this at your own risk, it worked fine on my version of WHM/cPanel/CentOS.