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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.

The potential deal would be one of the largest worldwide in the past decade, and rival Vodafone’s $181 billion takeover of the German cellphone operator MannesMann in 2000.

Analysts say Verizon’s purchase of the 45 percent of Verizon Wireless it does not already own would help the company to dictate strategy as it looks to invest billions of dollars in high-speed data infrastructure.

Verizon is still the No. 1 cellphone 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 to little avail — AT&T and Verizon still account for two-thirds of overall subscribers.

The wireless business, one of the most lucrative in the world, is crucial to the U.S. economy. Worldwide, the wireless industry, already worth $1.6 trillion, is expected to become a multitrillion-dollar market in the next decade, said Chetan Sharma, an independent telecom analyst that does consulting for carriers.

With already 10 billion connections worldwide, the number of cellular subscriptions is on track to outgrow the human population.

The deal would unlikely have any meaningful effect on Verizon customers.

In theory, having complete ownership of the wireless venture would allow Verizon to integrate the two businesses more tightly, which might lead to better deals on bundles with wireline and wireless products, said Jan Dawson, a telecom analyst for Ovum. However, Verizon and Vodafone already have been doing combined marketing for years, he said.

“The impact from a consumer perspective will be negligible,” Dawson said.

For Vodafone, the world’s second-largest cellphone 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 also would 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.’’

One of the biggest hurdles to the potential deal is the large tax bill Vodafone would have to pay to dispose of its holding in Verizon Wireless. Earlier this year, however, Verizon said it could structure any potential transaction to limit Vodafone’s tax liabilities.

Analysts said any prospective deal would likely involve a cash-and-stock offer that would give Vodafone roughly a 30 percent stake in Verizon.

Vodafone’s chief executive, Vittorio Colao, has previously said he was open to selling the holding in Verizon Wireless, though the company only confirmed on Thursday it was in talks about a potential deal.

The prospective disposal could be announced as soon as early next week, according to a source with knowledge of the matter, who spoke on the condition of anonymity because he was not authorized to speak publicly.

“There is no certainty that an agreement will be reached,” Vodafone said in a statement.

If Verizon and Vodafone were to come to an agreement, each company would have to take on a different outlook for the American phone business, said Craig Moffett, an analyst for Moffett Research.

To justify what would be one of the largest deals in history, Verizon would have to be confident that the growth of Verizon Wireless would remain consistently strong, he said.

By contrast, Vodafone would have to believe that the American wireless business was stagnating and that Verizon Wireless could not grow much more.

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

At least initially, investors seem to support the news. Shares in Vodafone closed up 8 percent in trading in London on Thursday.

Its stock price, however, has fallen around 40 percent since the Verizon Wireless partnership was established in 1999.

Verizon’s shareholders, too, seemed enthusiastic about the prospect of a deal. Its shares closed 2.7 percent higher Thursday, even though it would have to pay billions, and borrow billions, to make the purchase happen.