The 22-year-old was last seen on Saturday night in the New Zealand city, where she had been staying in a hostel.
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".
A 20-strong team of investigators has been scouring through CCTV footage to piece together her final movements before she disappeared.
Det Insp Scott Beard of the Auckland city police appealed for anyone who had been in or near the City Life Hotel at the time of Millane's disappearance to contact police.
Mr Millane will speak to members of the media this afternoon at 4pm, at a press conference at Auckland Central Police Station.
"Grace is a lovely, outgoing, fun-loving, family-oriented daughter", he said.
Grace's father, Mr David Millane arrived in the country just after 7am.
The University of Lincoln graduate, who planned to travel around the world, had two mobile phones with her in New Zealand but calls to both are going through to voicemail. At this point, we hold grave fears for her safety. "She's usually in daily contact with family on social media", he said. We last had contact with Grace on Saturday, December 1 and as a family we have been extremely concerned for her welfare.
"It is a very hard at this time to fully describe the range of emotions we are going through".
But according to a Facebook page set up by her family she never made it back to her room on Saturday night. 'A large part of that focus has been around CCTV footage around Auckland'. Her birthday was on Sunday but she didn't contact her family.
He said: "Police have identified this man and he has been spoken to".
"We have a couple of phones from the backpackers but we believe she may have another one or two so we're keeping an open mind", he said.
DI Beard would not confirm reports she had been using the dating app Tinder while in Auckland.
"We are conducting a scene examination at this address".
"It has now been six days since Grace was last seen", Beard said.
Anyone with information about the missing items has been advised to contact 0800 676 255.