"Mr Lam is an overstayer and a priority deportation case, however Immigration New Zealand has been unable to obtain travel documentation for him so far," said the spokesman.
"Government officials have discussed this issue with their counterparts in the US and Vietnam. We will continue to look for a solution, which may involve further work with the US and Vietnam governments."
Labour immigration spokeswoman Darien Fenton said Lam's reckless offending made him a potential danger to others.
"There needs to be high-level discussions to try and figure out a resolution. It's got to be minister-to-minister level otherwise it won't be resolved."
She said immigration officials were in a difficult position. "You just can't throw people on to planes and say goodbye."
A detailed account of Lam's criminal record was released to the Herald on Sunday by Immigration New Zealand.
He served just over four years of a 13-year sentence for importation and possession of 10kg of heroin.
During 2004 he was convicted in the North Shore District Court of theft, cannabis possession, and careless driving and fined between $100 and $200 on each charge.
Lam went on to commit another theft and was recalled to prison in August 2005, serving a further three years.
Since then, he has been convicted of four drink-drive and careless driving charges.
He was been granted limited purpose visas to enable him to work and to support himself as a painter decorator.
When the Herald on Sunday tracked him down to a Pakuranga address, Lam said he wanted to leave the country but was unable to.
Lam's family live in the United States, where he also has a criminal record.
