Two solutions come to mind.
The first is a simple frame/iFrame on the Linux website of your client enclosing you ASP.NET www.yourwindowsserver.com/clients/clientXYZ/MyPage.
This is simple to set up, but for the more tech savvy visitors of your clients website it will be visible that they're redirected to a second server.
The second alternative would be to configure part of the of the Linux website as reverse proxy. This has the advantage that to the vistors it will appear seamless as they only make connection to www.ClientDomainLinux.com.
The webserver on www.ClientDomainLinux.com will work as a HTTP proxy and take requests for /MyMvcController/MyPage, convert that incoming request to an outgoing request to www.yourwindowsserver.com/clients/clientXYZ/MyPage, receive the reply and will retransmit that reply to the websites visitor.
Linux webservers typically use an Apache webserver. A reverse proxy configuration either uses mod_proxy or alternatively mod_rewrite and this is typically something an administrator will need to configure.
# somewhere in the main apache configuration file
LoadModule proxy_module modules/mod_proxy.so
LoadModule proxy_http_module modules/mod_proxy_http.so
# either in the virtual host definition for www.ClientDomainLinux.com or possibly .htaccess
ProxyPass /MyMvcController/MyPage http://www.yourwindowsserver.com/clients/clientXYZ/MyPage
ProxyPassReverse /MyMvcController/MyPage http://www.yourwindowsserver.com/clients/clientXYZ/MyPage
The alternative with a rewrite rule would look something like the following:
# somewhere in the main apache configuration file
LoadModule proxy_module modules/mod_proxy.so
LoadModule proxy_http_module modules/mod_proxy_http.so
LoadModule rewrite_module modules/mod_rewrite.so
# either in the virtual host definition for www.ClientDomainLinux.com or possibly .htaccess
RewriteRule ^/MyMvcController/MyPage(.*) http://otherhost/otherpath$1 [P]
http://httpd.apache.org/docs/2.4/mod/mod_rewrite.html
http://httpd.apache.org/docs/2.4/mod/mod_proxy.html