I have been looking for an FTP client to transfer files between two server's the closest i got was FTP Rush, i am able to log into FTP account of both my Linux servers using FTP Rush but i am not able to transfer files. Can some one suggest some FTP client to do this.
This is an antiquated way of doing things. What you're using is FXP, which is a not very widely supported extension to FTP, to do a server to server transfer. Not a lot of clients support this because it makes more sense to either remote into one of the servers and initiate the transfer, use rsync, or automate via scripting. It is also a potential security threat. I would also imagine that FXP's lack of support extends to the server-side and you may not be able to get it work with some ftp servers.
FWIW, it seems like FlashFXP and FTPRush are the leaders in FXP transfers. If neither work then your server doesn't support FXP or needs special firewall rules to make it work (my understanding is that none of this works in passive mode and explicit open ports need to be made in the firewall).
Also, this is important (from wiki article)
"Enabling FXP support can make a server vulnerable to an exploit known as FTP bounce. As a result of this, FTP server software often has FXP disabled by default."
First off - see if there's any way you can avouid using FTP - it's famously insecure and a PITA for securing NAT traversals.
What's wrong with 'ftp' which comes with every Linux and Unix distro I've ever come across, and even MSWindows?
A quick google, and the only 'ftp rush' I found was something which only runs on Microsoft. Since the only OS you mention in your q is Linux, it might be helpful if we knew what OS you intend running the client on.
There's lot's more - wget and curl will both handle FTP as well as HTTP. Not to mention lots of gui type apps on Linux/Unix and MSWindows. Indeed, MSWindows Explorer will talk to FTP servers.
What do you want this program to provide? Scripting? A nice UI? Something else?
Sounds like it might be your network config that's the problem here - not the program.
Transferring directly from one FTP server to another is technically possible but subject to issues with firewalls and supported by very few client applications.
If you have a fast enough connection you could use an FTP client that fully supports drag-and-drop (windows explorer does this natively) and transfer the files this way, but I assume this is exactly what you are trying to avoid (as you end up downloading the content to your location then pushing it up to the second server).
You don't specify what OS is running on the servers and what sort of access you have, so I can't give advice specific to your circumstances. If at least one of the servers is accessible via SSH then you can login that way and FTP from/to the other server using the command line tools. If both are accessible via SSH then you can almost certainly use the much better SCP/SFTP protocols and may even be able to use rsync which has added advantages.
Edit: Your question didn't stipulate other people (presumably relatively non-techie users) being able to transfer files between the servers. You could install an http-interface based file manager such as http://extplorer.sourceforge.net/ (there seem to be quite a few available, that one I have bookmarked to look at when I have time to play with such things but I've not tested it or any of the others yet myself) for your users to use. extplorer apparently supports FTP sites directly, but if you pick one that doesn't you could link the servers' filesystems via a mounts over a VPN and allow your users to manage files between the two that way.