"The Greeks look to have voted no, souring global risk sentiment," ANZ Bank New Zealand senior economist Philip Borkin and senior FX strategist Sam Tuck said in a note.
"Sentiment is driving 'risk' currencies lower and safe-havens higher."
Concern about a slump in Chinese equities is also weighing on the New Zealand and Australian dollars, ANZ said. Chinese authorities at the weekend announced that the top 21 brokerage firms were to form a fund to support blue-chip stocks.
The kiwi will probably trade between 66 US cents and 67.50 cents today, ANZ said.
In New Zealand today, data on new vehicle registrations for June may be released.
The Motor Industry Association provisional data shows it may be the biggest June month on record for new vehicles, with close to 13,000 registrations.
MIA records show only four occasions when monthly sales have previously broken 13,000.
The New Zealand dollar advanced to 89.23 Australian cents from 88.58 cents on Friday ahead of tomorrow's Reserve Bank of Australia meeting.
The local currency touched 80.84 yen, its lowest level since November 2013, as investors favoured the perceived safety of the yen. It was trading at 81.09 yen at 8am from 82.75 yen on Friday.
The kiwi slumped to 42.69 British pence, its lowest level since September 2009, and was trading at 42.82 pence at 8am from 43.06 pence on Friday. It was little changed at 60.53 euro cents from 60.58 cents on Friday.