NEW DELHI—Washington has hit an unlikely roadblock in its extraordinary global push to sideline China’s Huawei Technologies Co.: the world’s biggest democracy, India.

Policy makers and telecommunications firms here are so far largely unpersuaded by U.S. warnings that using Huawei’s equipment to upgrade India’s telecom networks presents a major cybersecurity threat, according to more than a dozen government officials and industry executives. Many argue that any such risk is outweighed by Huawei’s cut-rate prices and technological...