On Tuesday Twyford said that Cabinet would "soon" decide on its preferred delivery partner - either the NZ Transport Agency or NZ Infra, the joint venture between the New Zealand Super Fund and Canada's CDPQ Infra group.
But asked about the project, Peters said there had been massive cost blowouts.
"It's not going to happen in the immediate term," he said.
"We've always been for heavy rail around this country. Our programme is on target, as you know, and light rail has been suspended in terms of planning for the immediate future."
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Last year Peters also raised concerns about cost blowouts for the project, which has been estimated to cost $6 billion.
But NZ Infra director Will Goodwin dismissed a costs blowout - though he wouldn't be drawn on specifics.
"We've seen speculation of costings blowing out to as much as $10b. While we cannot get into specifics about what the cost of our proposal will be, we've previously said this speculation is substantially wide of the mark.
"Costs incurred to date are commercially sensitive and confidential."
He would not say how much money NZ Infra had sunk into it so far, but he remained committed to the project and supported Twyford's view that Cabinet would look at the issue soon.
"We understand and accept Government has been totally focused on its response to the Covid crisis. The Prime Minister has said the project remains on the Government's agenda and we understand Cabinet will consider it shortly.
"NZ Infra remains committed to delivering light rail from the city centre to Māngere and the airport."
The Ministry of Transport said the project would help with the country's post-Covid economic recovery.
"The preferred delivery partner, when chosen, will run a procurement process to select those it will partner with in building and operating the project.
"Given the size of the project, the initial preparatory phases alone require investing significant resources and will help with the country's economic recovery, even before construction starts."
Public consultation and a consenting process would have to happen after the delivery partner was selected.
"The project will revolutionise transport in Auckland and represents a substantial investment in future-proofing the region's growth and sustainably increasing productivity," the ministry said.
Official papers and a statement from the NZTA previously showed the task was more difficult than Labour envisaged and a business case for the CBD to airport line was still being worked on months after it was due in December 2018.
Since then there has been ongoing back-and-forth about who will build the scheme. The two bids have been analysed but a winner has not yet been chosen.