At the same time he announced plans to sell between 1000 and 2000 state houses a year to community housing providers, promising to keep at least 60,000 houses in Housing NZ by 2017.
Social Housing Minister Paula Bennett said in the same press statement: "This won't reduce the number of social housing places available - indeed these will increase by 3000 by 2017-18."
But the Budget documents show that the number of social housing places actually dropped to 60,260 by the end of last month, mainly because Housing NZ and other providers have been selling and demolishing existing houses in redevelopment areas such as Tamaki at a faster rate than they have been building new ones.
The number of "social housing" places is actually lower than the number of state houses (67,041 at the end of March) because the term "social housing" refers only to houses where the tenants' incomes are low enough to qualify for income-related rents set at 25 per cent of their incomes. The Government pays Housing NZ and other providers a subsidy to top up the income they receive to local market rents.
This year's Budget actually allocates an extra $22.3 million for income-related rent subsidies in the financial year that started this month, $58m in 2017-18, $78.5m in 2018-19 and $41.2m a year from 2019-20 onwards - apparently matching the $40m a year that Key said it would cost to fund 65,000 places.
But the pre-Budget report, dated March 24, says this will increase the number of social housing places in 2017-18 by only 750, "from the current forecast of 63,392 to more than 64,100".
"As discussed with ministers in recent months, this departs from the attempt to reach a target of 65,000 social housing places," the report says.
Large parts of the report have been deleted from the public version, but it appears that the change may be due to higher costs for each place. The report says the assumptions have been changed to give the housing providers a 6.1 per cent return on investment for capital improvements.
Mr Twyford said Housing NZ told MPs that the number of social housing places would increase to 64,172 by 2019-20, including an increase in Auckland from 29,700 in the current year to 31,138.
He said Labour would budget an extra $11m for every net increase of 1000 state houses. It has promised a net increase of 1000 a year, costing an extra $33m a year by the third year.
Bennett said the target of 65,000 had not been reduced.
"As the Treasury Budget documents point out, the precise number of places that can be funded depend on the deals the Ministry of Social Development is able to negotiate and therefore the price the Government pays for its rents," she said.
"It comes down to a lack of supply - Auckland doesn't have enough properties to put people in, that's a challenge which is why we have a comprehensive plan to speed up the supply pipeline of social housing.
"This includes Housing NZ spending $2 billion to build around 4000 houses over the next three years, building modular housing to help us speed up our redevelopments, working with community housing providers to grow their supply and spending more on emergency housing."