I'm a z/OS Systems Programmer and was looking for a solution for this. After some extensive reseach I found that there is a third party product that is doing exactly what you need.
Quote from their webpage:
"Type80 Syslog for z/OS enables extension of all mainframe console messages and write-to-operator messages to be routed to external log retention servers using the standard TCP/IP Syslog protocol".
More info here:
http://www.type80.com/products_syslog.htm
I'm still trying to find something that is free.
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