c.2013 New York Times News Service

c.2013 New York Times News Service

LONDON — Verizon Communications and Vodafone moved one step closer Thursday to parting ways.

Vodafone, the British telecommunications giant, confirmed that it was in talks to sell to Verizon its 45 percent stake in Verizon Wireless.

The future of Verizon Wireless had been in the balance in recent months after speculation surfaced that Vodafone would sell its holding in the joint venture, a deal that analysts said could be worth up to $125 billion.

Verizon is still the No. 1 cell phone carrier in the United States by market share, but it faces formidable competition from AT&T, the No. 2 carrier. The smaller carriers, Sprint and T-Mobile USA, offer lower-cost phone and data plans to try to compete, but AT&T and Verizon still account for two-thirds of overall subscribers.

For Verizon, the challenge will be proving to investors that there will be financial benefits to having complete ownership of its wireless unit. The company already owns 55 percent, so assuming full ownership will not allow it to gain new assets as a typical acquisition would. The deal would unlikely have any meaningful effect on Verizon customers.

For Vodafone, the world’s second-largest cell phone operator behind China Mobile, an influx of cash would allow it to strengthen its core European operations, which have struggled because of the Continent’s financial woes. It would also allow Vodafone’s investors to benefit through share buybacks.

“Vodafone investors are expecting a fairly material payout,” said Paul Marsch, an analyst at Berenberg Bank in London. ‘‘They have been waiting for a very long time.’’

To justify what would be one of the largest deals in history, Verizon would have to be confident that the growth of Verizon Wireless will remain consistently strong, said Craig Moffett, an analyst for Moffett Research. By contrast, Vodafone would have to believe that the American wireless business is stagnating and that Verizon Wireless cannot grow much more.

“For investors, the pertinent question is therefore: which outlook do you believe?” Moffett said.