The decision means the next Cup defence will head abroad, despite the $99 million cash and support offer tabled by the Government and Auckland Council.
"Representatives of the Team New Zealand Board have informed the Crown-Council negotiators that the offer is not sufficient," America's Cup Minister Stuart Nash said.
"The offer expires today, 16 June. From tomorrow, Team New Zealand is now free to seek support from other partners."
Nash said the Government wished the syndicate well.
"We want the next America's Cup raced in New Zealand. Tens of thousands of diehard Kiwi fans who turned out to support the historic defence of the Cup in March want it raced in New Zealand. It is disappointing we were unable to reach agreement within the exclusive negotiation period.
"At the end of the day, the America's Cup is a global commercial operation. It is an international business as much as a sporting contest.
"The team is now free to look to commercial sponsors, private supporters, or other avenues to bankroll the operation."
Nash said Team NZ was a "world-class" operation.
"I know they will fly the flag for their home nation in the 37th America's Cup, whether it is held in Auckland, or taken offshore.
"Any subsequent request for government support would have to be considered by Cabinet and I cannot confirm whether the Crown would consider making another offer."
Earlier: PM - we still wanted to host the Cup
Prime Minister Jacinda Ardern had reiterated the Government still wanted New Zealand to host the next America's Cup, amid signs the defence is heading overseas.
Ardern told Radio Hauraki this morning: "Our view is that we want it to be hosted here.
"We've put our best foot forward, but there's also limits to what we can do," she said.
"Because, you know, this is taxpayer-funding, ultimately - so it's got to always be to a benefit to New Zealand."
Ardern said there was therefore a point where one could not justify the amount of money going in.
"We've tried to make that judgment about where that line is. We've put our best foot forward.
"Really now, it's in their court - that's how I see it."
Speaking to The Hits, Ardern noted that there had not been a time when New Zealand had held the Cup and not hosted it.
It also emerged this morning that a surprise overseas venue had been tipped to bid for 2024 America's Cup.
Team New Zealand CEO Grant Dalton on Tuesday night told more than 400 members of the Royal New Zealand Yacht Squadron (RNZYS) that it was extremely unlikely that an agreement with the Government to stage the next event on Kiwi waters would be reached, after more than three months of negotiations had failed.
Dalton was addressing RNZYS members at the club's headquarters on Team NZ's reported plans to take the event offshore.
The Kiwi syndicate retained the Auld Mug by beating Italian outfit Luna Rossa in the Cup Match in March.
