Detective Inspector Scott Beard and David Millane spoke to media this afternoon.
Beard said police held grave fears for the 22-year-old tourist's safety, and the investigation was progressing rapidly.
The last known sighting of Grace was at 9.41pm on Saturday at the CIty Life Hotel. She was with a man who had since been identified and spoken to by police.
An apartment at the hotel was a place of interest and is the subject of a scene examination.
The New Zealand Herald spoke to the manager of the hotel, who said he did not know anything.
A couple who have been living at the hotel for the last seven months said they were concerned for the girl and her family but had not seen or heard anything.
Detective Inspector Scott Beard appealed for anyone who could provide information to contact police on 0800 676 255.
"At present we have spoken to a person and we still have no evidence of foul play."
David Millane, fighting back tears, struggled to speak as he described his "lovely, outgoing, fun-loving, family-oriented daughter".
"Grace has never been out of contact for this long. She's usually in daily contact with her mother, myself, her two brothers ... and on social media."
His daughter had started her year-long OE in Peru and had told her family how much she was looking forward to moving on to New Zealand.
"She'd been bombarding us with numerous photos and messages of her adventures. We're all extremely upset and it's difficult to describe the wave of emotions we're going through."
He appealed for people with information to contact police - no detail was too small, Millane said.
The millionaire property developer, who was met by police when he flew into New Zealand at 7am today, was last in contact with his daughter on Saturday.
More than 25 police staff were working on the investigation after the young woman from Essex disappeared in Auckland after Saturday night.
Beard said it was understood Grace had a New Zealand and an English cellphone.
In response to questions, Beard said she could have more cellphones.
He would not be drawn on whether Grace knew the man prior to their meeting at the hotel.
Police would continue to go through CCTV footage over the weekend, he said.
Who is Grace Millane?
Grace Millane was last seen on Saturday night in downtown Auckland, the day before her 22nd birthday.
Her family raised the alarm two days later, after she stopped responding to their messages. Prior to then she had been in contact daily.
The "bubbly", "outgoing" artist and advertising graduate from Essex, in the southeast of England, was travelling solo for the first time.
Her brother Michael told the Herald she was six weeks into a year-long overseas adventure.
She started her trip in South America, spending four weeks travelling as part of a group itinerary.
A month ago she hiked with the group to the Inca citadel Machu Picchu in Peru.
Once she arrived in New Zealand about two weeks ago she travelled around the upper North Island.
