Millane's brother, Michael Millane, posted an appeal for help on Facebook.
Detective Inspector Scott Beard, of Auckland City Police, said Ms Millane's disappearance was "out of character" and that her family were understandably "distraught", with her father already en route to New Zealand to join the search. "Please share and contact if you have any information regarding her whereabouts".
Can everyone please share this, we haven't heard from my sister since Saturday is now travelling in New Zealand and was last seen in Auckland.
Police are asking anyone who may have seen her or who knows where she is to get in touch.
Base Backpackers is located on Queen St in Auckland's CBD.
She was wearing a black dress and what are believed to be white Converse trainers.
Before New Zealand she had spent four weeks travelling in South America.
"The longer this goes on the more concerning it is".
Michael Millane pleaded for his sister to contact the family.
The family put the call out there because they were anxious for her safety and wanted "to make sure she's alright".
The radio silence was "unusual" for Grace, Declan Millane said. She is approximately 5ft 6in (168cm) in height and 58kg.
She was just sight-seeing and travelling around New Zealand.
There are more than 20 staff working on her case, he added.
Base Backpackers were approached by Stuffon Wednesday but declined to comment.
A Foreign Office spokesman said: "We are in contact with New Zealand police and providing consular support after a British woman was reported missing in Auckland".