Ms Millane's brother, Michael, said she had travelled to Peru as part of an organised group for six weeks but had not travelled on her own - until she arrived in New Zealand two weeks ago.
A spokesman for the Foreign Office said: "We are providing consular support to the family in the United Kingdom, and working with the New Zealand Police in relation to reports of a missing British national in New Zealand". "At this stage we're still hopeful that we'll find Grace".
Detective Inspector Scott Beard said officers had spent the past 24 hours trying to piece together her last known movements, who she has associated with and the circumstances of her disappearance.
Meanwhile, her father had chose to fly to Auckland from Essex, in southeast England, on Wednesday night (England time) and was due to arrive in New Zealand on Friday.
Her older brother Michael Millane told United Kingdom news website Echo that the family are finding the uncertainty extremely hard.
Her family, from Wickford, near Basildon, said they spoke to her "most days", describing her silence as "completely out of character".
During that time she had been in touch with her family daily through WhatsApp.
Ms Millane's father, David Millane, 60, a millionaire property developer, is in the process of flying to New Zealand to help with the search.
Thousands of people have shared the post.
South Korean President says visit from North Korea's Kim "a possibility"
Trade is also likely to feature, following New Zealand and South Korea signing a free trade agreement in 2015. Moon has met Kim three times this year.
Base Backpackers is located on Queen St in Auckland's CBD.
She has brown hair and brown eyes and is approximately 5ft 6in (168cm) in height, and 58 kg.
A missing persons report was filed on Wednesday and "initial investigations are underway", New Zealand Police told Yahoo7.
Tony McLeod, consular manager at the British High Commission in Wellington, said the commission were providing consular assistance to the family back in the UK.
'We are unable to make further comment while police are investigating the matter'.
She was wearing a black dress and what was believed to be white Converse trainers.
"There have been a few tips and we will be working through them".
Her family in the United Kingdom raised the alarm on Monday and Detective Inspector Scott Beard said her disappearance was "out of character". "Please share and contact if you have any information regarding her whereabouts".
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