“And waiting on Softbank to finance the deal?” Mr. Legere wrote.

Mr. Ergen said publicly this week that several potential lenders had emerged to help his company buy assets, including JPMorgan Chase and SoftBank.

Sprint and T-Mobile, the third- and fourth-largest wireless companies, announced their latest merger plans in April 2018. The carriers promised their union would allow them to combine resources and bring the next generation of wireless broadband, known as 5G, for fifth generation, to rural America. They would have a combined 80 million United States subscribers.

The Justice Department announced its approval of the deal in July, citing the creation of a fourth and new competitor in Dish, which would buy assets from Sprint and T-Mobile to become a telecom company. In a parallel review, the chairman of the Federal Communications Commission announced it planned to approve the deal weeks later.

The merger is being challenged in court by several states and cannot close until that lawsuit is resolved. State attorneys general in New York and California are unconvinced that Dish will provide true market competition.

“Dish is a struggling satellite TV firm with no experience running a mobile wireless business — and no current mobile wireless business,” Paula Blizzard, California’s deputy attorney general, said on a call with journalists this month. “We cannot count on Dish one day in the future somehow growing into a viable wireless company equal to Sprint’s reach today.”

Mr. Delrahim was pressured to block the merger throughout the department’s review. Several Democratic lawmakers, consumer groups and state attorneys general said the deal would harm consumers by reducing the number of national wireless carriers to three from four. The reduction in competition would most likely lead to higher consumer wireless bills, the critics warned.

To salvage the deal, the companies came up with a solution: bring in Dish Network to buy some of their wireless assets to form another competitor and maintain four national mobile carriers.