Mr. Hussain said that a single client had hired the entire call center after the state stopped its surveys. He said that the deal had been struck by his boss, the company’s managing director, and he claimed not to know the identity of the client or the specifics of the contract. “We have rented out the manpower and the infrastructure to another company,” he said. “How and what they are doing is not my business.”

Alex Paul Menon, the chief executive of the Chhattisgarh Infotech Promotion Society, which runs the SKY program, said India was keen to spread digital literacy. In a state like Chhattisgarh, where simple access was a problem, mobile phones were an obvious first step. In 2016, Mr. Menon’s agency began to seek a mobile carrier to build towers in sparsely populated areas.

The contract, which will be worth more than $100 million when completed, went to Reliance Jio, a fast-growing carrier run by a billionaire supporter of Mr. Modi, Mukesh Ambani, and to Micromax, a financially struggling Indian phone maker. The first batch of phones, which each cost the state about 2,500 rupees, or $35, was distributed over the summer.

“It’s not like someone woke up this morning and said, ‘Oh, this is going to get votes,’” Mr. Menon said. “It’s nation building.” He said his agency kept a tight leash on user data and had not shared it with anyone.

Mr. Menon said a committee of top state officials was appointed earlier this year to choose apps for the phone and decided to load the Singh and Modi political apps along with government apps that help users tap national and state services.

The state says that its research shows that customers are using the phones to speak with family members, surf the web and send WhatsApp messages. But about a dozen recipients interviewed last week in Rajnandgaon said they were barely using the phones. They complained of poor battery life, apps that crash, and a tiny monthly allocation of free data — concerns that also popped up in the state surveys conducted through October. Micromax and Jio said no one was available to answer questions about the program.