“We have an existing agreement,” William Duma, the state investment minister for the Pacific island nation, told reporters. Huawei has already completed 60 percent of the $200 million project, which is financed by the Export-Import Bank of China. Mr. Duma said that made the 11th-hour effort from the Western powers “a bit patronizing.”

The competing offer comes as concerns have grown in Australia and allied nations about China’s investments in the resource-rich island countries of the Pacific Ocean. This year, Australia agreed to fund an undersea communications cable linking Sydney with the Solomon Islands, following a similar deal with Papua New Guinea.

Australian officials were worried that if the project went to Huawei, as the Solomon Islands had agreed in 2016, the Chinese company might gain access to Australia’s internet infrastructure. In 2012, Huawei was barred out of security concerns from bidding on projects for Australia’s national broadband network. The company has, however, sold equipment to Australian cellular operators.

Like Australia, New Zealand has been grappling with a series of scandals related to Chinese influence. Last month, a New Zealand lawmaker was accused of trying to hide a campaign contribution from a businessman with ties to the Chinese Communist Party. Another lawmaker was revealed last year to have been a member of the Communist Party in China who taught English to spies there.

More recently, agents of Beijing were blamed for a burglary targeting a New Zealand professor who studies the Chinese Communist Party’s influence in Western countries.

Despite those concerns, New Zealand has been relatively tactful in dealing with China, experts say. As a result, its two-way trade with China has more than tripled in the past decade, according to government statistics, and the bilateral relationship has become one of New Zealand’s most important.

Last year, New Zealand became the first advanced economy to join China’s Belt and Road Initiative, an ambitious infrastructure program stretching through much of Asia that is widely seen as an effort to extend Beijing’s political influence.