CANBERRA, Australia—The U.S. is vying to build an internet network in Papua New Guinea to prevent a Chinese telecom firm from doing so, a senior U.S. official said, signaling a widening effort to counter Beijing’s influence in a region Western allies have dominated since World War II.

The acting U.S. ambassador to Australia, James Carouso, said Friday that the U.S., Japan and Australia were preparing to counter a $200 million contract that Papua New Guinea recently awarded to China’s Huawei Technologies Co. to build a cable...