Is client-side port/connection sharing available?

Hello.  We have a scenario where the client machine has a very limited number of open ports for applications to use.  I believe that each client database connection on the *clinet* machine will use a different port to connect to the server port (1521). Is there a feature in TNS or another middleware component that will allow multiple processes on a client machine to use a very limited number of ports for database connections?  Here's hoping we don't have to significantly rewrite the client application!

We have an 11.2 RAC with client applications that connect primarily through python/cx_oracle (with a few OCI).