John Minchillo/Associated Press

With nearly 100 million Americans using Verizon Wireless for their cellphone service, it may not seem obvious that almost half of Verizon is owned by a company overseas.

That could soon change. Vodafone, the British telecommunications giant, has confirmed that it is in talks to sell to Verizon Communications its 45 percent stake in Verizon Wireless, a deal that analysts say could be worth at least $125 billion.

For Verizon, this has been a long-sought deal, one that would rank among the biggest purchases in history. With complete ownership of its wireless business, the company would be able to shift from receiving dividends to being able to fully incorporate all of its profit. And it will have full control over what it does with that profit, especially as advertisers, content distributors and wireless carriers increasingly use smartphones, mobile data and information services.

In theory, having complete ownership of the wireless venture would allow Verizon to integrate its businesses more tightly, which might lead to better deals on bundles with wireline and wireless products.

At least in the short term, Verizon customers would be unlikely to see much difference in their service. “The impact from a consumer perspective will be negligible,” said Jan Dawson, a telecom analyst for Ovum.

But it is clear why Verizon would want complete ownership of its wireless division. The lucrative 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 who does consulting for carriers.

With 10 billion connections worldwide, the number of cellular subscriptions is on track to outgrow the human population. That is because in addition to cellphones, many other devices, like tablets, video game devices and even home security systems, are now all relying on cell towers to deliver information, entertainment and media to consumers.

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.

Verizon’s core strategy has been to invest more in network infrastructure to attract customers with the best technology. For instance, it is leading the industrywide race in building a faster fourth-generation wireless network, called LTE. Verizon has LTE covering 500 markets; AT&T has LTE in about 370 markets.

Verizon, one of the last so-called Baby Bells that trace their origins to the breakup of AT&T, has banked on wireless as a central part of its future. Today, a majority of Verizon’s revenue comes from its wireless business. In the second quarter this year, Verizon Wireless brought in $20 billion of Verizon’s total $29.8 billion in revenue.

For the most part, Vodafone has been an unseen partner all these years, happily taking in its cut of the profits via the dividend that its 45 percent stake in the wireless venture has provided. The relationship is one that dates back to the telecom mergers in the 1990s. Bell Atlantic had merged with GTE to become Verizon Communications. Vodafone had acquired AirTouch, an American wireless company. Verizon merged its wireless business with Vodafone’s acquired assets of AirTouch so that combined, they could become a bigger player.

For years, there was speculation that Vodafone would want to get out of the business so it could concentrate on its European operations, but the right price and tax implications of any deal had always held up any definitive agreement.

This year, however, Verizon said it could structure any potential transaction to limit Vodafone’s tax liabilities. Despite speculation this year about a potential deal, no acquisition for Verizon Wireless materialized, and Verizon said in April that it did not have plans to merge or make an offer for all of Vodafone.

Analysts said any prospective deal now 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 that he was open to selling the holding in Verizon Wireless, though the company confirmed only on Thursday that it was in talks about a potential deal. A final deal could be announced as early next week, according to a person with knowledge of the matter, who spoke on the condition of anonymity.

If Verizon and Vodafone were to reach 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 the billions it would have to pay, Verizon would have to be confident that the growth of Verizon Wireless would remain strong, he said. 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?” Mr. Moffett said in a research note.

At least initially, investors seemed to support the news of the talks. 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 were up about 2.7 percent, even though it would have to pay, and borrow, billions, to make the purchase happen.

A Vodafone spokesman declined to comment further. A representative for Verizon was not immediately available for comment.

The potential deal would be one of the biggest in the last two decades, trailing only Vodafone’s $202.8 billion takeover of the German cellphone operator Mannesmann in 2000 and the $181.6 billion merger of AOL and Time Warner in 2001, according to Thomson Reuters. (Both figures include the assumption of debt.) And between fees for advising the two companies and for lending money to support the deal, a transaction would be a bonanza for advisers to Verizon and Vodafone.

For Vodafone, the world’s second-largest cellphone operator behind China Mobile, an influx of cash would let it 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.”

The healthy earnings at Verizon Wireless stand in contrast to those of Vodafone, which has experienced anemic growth in Europe.

Vodafone reported a 3.5 percent fall, to $15.7 billion, in its so-called joint service revenue — a measure of its continuing services that does not include handset sales — in the three months that ended June 30.

The decline was the fourth consecutive quarterly drop in revenue, and highlighted persisting weaknesses in the company’s crucial German and British markets.

“Vodafone faces a strategic challenge in its European business,” said Mr. Marsch of Berenberg Bank. By selling its stake in Verizon Wireless, Vodafone could receive a large war chest to finance potential acquisitions and renewed investment in so-called fourth-generation wireless networks.

Vodafone remains either the No. 1 or No. 2 operator in eight of its nine Western European markets, but it is facing increased competition from cable companies like John C. Malone’s Liberty Global, which are offering bundled service packages that include high-speed broadband and telephone services.

Despite its strong market share in mobile phone offerings, analysts say Vodafone needs to improve its operations, particularly in cable, to defend against the ramped-up investment plans of its rivals.

“Vodafone needs to improve its bundled services for its customers,” said Gyanee Dewnarain, a research director at the analyst firm Gartner in London. “People are looking for more value for their money.”

Michael J. de la Merced contributed reporting.