That plan could be foiled by Niel, whose French wireless provider, Iliad SA, approached T-Mobile’s board with a $33-a- share bid for control of the company. Deutsche Telekom, which owns two-thirds of T-Mobile, considered the offer less competitive than the proposal of about $40 a share that Son is said to be preparing, according to people familiar with the matter.

Then there’s another billionaire: Dish Network Corp. Chairman Charlie Ergen, who has said he could go after T-Mobile if Sprint fails to buy it.

U.S. regulators may prefer a buyer like Iliad or Dish that would preserve a market with four national carriers, since Son would seek to combine T-Mobile and Sprint into a single company. Son has argued that a merger would help him gain the capital he needs to invest in the faster wireless Internet speeds that consumers crave.

Important Infrastructure

“The mobile Internet is the most important infrastructure for the 21st century,” Son told Charlie Rose in an interview earlier this year. “I would like to provide U.S. citizens the world’s No. 1 network.”

With his SoftBank already among Japan’s largest wireless carriers, Son is straddling two of the regions that are expected to dominate the mobile Internet. Cisco Systems Inc. forecasts that global data traffic over wireless networks will climb at a 61 percent annual clip to 15.9 trillion megabytes in 2018 -- the equivalent of about 4 trillion high-definition movies. North America and Asia will represent two-thirds of that traffic.

And while Asia will have the biggest share of traffic because of its larger population, North America will have a bigger share of advanced gadgets that require ever-faster download and upload speeds -- which let wireless carriers charge more. In North America, 93 percent of mobile devices and connections will be “smart” -- with advanced computing and multimedia capabilities -- by 2018, compared with 47 percent in Asia, Cisco said.

‘Jaw-Dropping’ Divergence

Americans have kept on paying a steady rate for wireless services as speeds get faster and new features like video are added to their plans. U.S. carriers got an average of $48.17 a month in revenue per user in the fourth quarter, compared with $32.51 in France, according to researcher Informa Plc.

Wireless revenue per capita rose 17 percent in the U.S. from the beginning of 2009 through the middle of 2013, compared with a 4 percent decline for the rest of the developed world, according to MoffettNathanson LLC.