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The 29 comprise 24 New Zealanders, two Australians and two Scots and a South African.
Many have already been named publicly.
Grey District Mayor Tony Kokshoorn has called for all the names to be released, saying there is no point in keeping them secret.
"I'm all for disclosure of the names to avoid speculation. Everyone knows locally -- it's not a secret and anyone could find out if they wanted to," he told NZPA.
There has been some anguish from the families over the release of the names, but Tasman area police commander Superintendent Gary Knowles said today he would speak with them at a conference tomorrow morning and explain why police thought they should be released.
The British Foreign Office has confirmed the two Scots are Peter Roger, 40, and Malcolm Campbell, 25, both New Zealand residents.
Mr Roger emigrated to New Zealand two years ago to be closer to his mother and sister, who live here, the Courier newspaper said.
He was formerly an offshore oil engineer.
The parents of Mr Campbell, from St Andrews in Fife, said they had to keep hoping for good news.
His father, Malcolm, told Scottish television: "We can't concentrate on anything, we can't sleep because it's difficult.
"Our prayers and thoughts go out to everybody who is going through this in New Zealand. We just keep hoping that everything will be fine."
Mr Campbell is due to marry his fiancee, Amanda Shields, 23, on December 18.
One of the Australians is Josh Ufer, 25, whose partner Rachelle Weaver is three months pregnant, the Sydney Morning Herald reported.
Ms Weaver, 23, of Middlemount in Queensland, said she would not give up hope that Ufer, a driller supervisor, would emerge alive.
"We want people to keep doing what they're doing, because we know they're doing the best they can," she said through her mother, Nancy Langley.
Mr Ufer was working for contracted Hunter Valley company Longwall Drilling. His parents, Joanne and Karl, and sister Kymberley, are understood to be on their way to Greymouth to join the vigil.
The name of the other Australian has not been confirmed but he is believed to have a wife and two children.
The New Zealanders include Grey District councillor Milton Osborne, West Coast rugby league representative Blair Sims and West Coast rugby representative Michael Monk.
Benjamin Rockhouse, whose brother Daniel was one of two miners to escape the gas explosion, is also one those missing.
Other miners named in media outlets today are Brendon Palmer, Alan Dixon, Zen Drew, Chris Duggan, Peter O'Neill, Terry Kitchin, Glen Cruse, Conrad Adams, Kane Nieper and Joseph Dunbar, who was reportedly on his first day at work in the mine.