In a tweet, the council said it was not yet known how long the outage would last.
It described the developing situation as a national outage and said calls to its Customer Service Agency landline were not connecting at present.
Due to a national outage, calls to our Customer Service Agency landline may not be connecting at present. In the meantime, you can still visit https://t.co/EQfsIMcppK, email us at dcc@dcc.govt.nz, or contact us through Twitter & Facebook.
Metlink Wellington said it was also experiencing issues with their phone lines and told customers to refer to the transport agency's app or website for information.
Mōrena Wellington, we are currently experiencing issues with our phone lines. We are working to fix this and will update when this matter has been resolved. Ngā mihi^JB
Vodafone spokeswoman Nicky Preston said some business customers across New Zealand had been affected by the technical issue that had hit this morning.
"We're urgently working to fix a technical issue impacting some corporate customers in New Zealand which is causing some calls to call centres to fail."
Preston said it was primarily affecting banks and utilities.
Teams were working to fix the fault and it was hoped lines would be restored soon.