GTM and VS with public and private adresses

are your server(LTM) and virtual servers in GTM configured with addresses and translations? If so, the monitors should be fine on internal addresses while still handing out external answers to queries. If you will have internal queries needing the internal address on same wideIP, you can do a couple different things. You can create an internal view of the names/ip's in question, then write an iRule to forward to bind if client ip addr is internal. Or, you could do it all in an iRule, though since GTM iRules don't support classes, the rule could get quite long if you have a lot of records.