Mr. Claure said that his primary job was to improve Sprint’s health as a stand-alone company. That progress has been slow, but has shown results: The telecom company reported $206 million in profit in the second quarter, the first such profit in three years. Much of that came from an expansive cost-cutting program that has shed nearly $4 billion over the past two years.

But Sprint still lost 39,000 consumers on traditional contracts in the quarter, despite offering a full year of unlimited data to new subscribers. The company’s main rivals all reported strong gains in new customers for the quarter.

Sprint has had to acknowledge that it would have a tough time thriving on its own. From the company’s perspective, combining with Charter would make sense.

Broadband providers have been eager to find other ways to drive up revenue, and wireless has been one avenue that cable operators have dived into. Charter and Comcast formed an alliance this spring to explore wireless opportunities. Already, Comcast has begun offering its subscribers mobile service through Verizon, with Charter expected to follow suit.

From Sprint’s perspective, a cable company would help give the telecom company the wired backbone it needs to upgrade its wireless service to 5G. While Verizon and AT&T have been investing billions into bolstering their networks, Sprint has had far fewer resources to do so.

“The potential additional synergies that you have in doing a strategic transaction will always be significantly better than having a stand-alone entity,” Mr. Claure said on the earnings call.

Few on Wall Street would discount the ambition of Mr. Son, who built SoftBank into a behemoth of Japanese business. The American-educated businessman founded SoftBank in 1981 and expanded its universe to include broadband, technology and more. Much of that arose out of the sheer will of Mr. Son, who threatened to light himself on fire in the offices of Japan’s telecom regulator at least once.