A BNZ spokeswoman said also demand for sterling had been significantly higher than usual and some customers had faced delays in receiving their currency orders this week.
"We're monitoring stocks and we do have pounds [available]," she said.
It's a similar story at Westpac, which is attributing heightened sterling demand to the upcoming school holidays as well as the post-Brexit slump.
"To ensure we meet customer demand we have ordered extra currency," a spokeswoman said, adding that Australian dollars were also in high demand for similar reasons.
The amount of foreign currency held at Westpac branches varied across different locations.
"While the vast majority of branches have had no issues, a couple of branches that have not usually carried a high volume of [pounds] have run short but have received new supply quickly," the spokeswoman said.
Many customers have been opening GBP foreign currency accounts as an alternative to cash if they are not planning to travel in the immediate future.
An ASB spokeswoman said that while some branches may run out of British pounds as a result of surging demand, the bank was still in a position to meet orders.
"Many customers have been opening GBP foreign currency accounts as an alternative to cash if they are not planning to travel in the immediate future," she said.
Britain's vote to leave the EU after 43 years sent the pound crashing around 12 per cent against the US dollar to a 31-year low, although it has gained back some ground over the course of this week.