It has confirmed it is discussing selling the stake to Verizon Communications for $130bn (£84bn).
The company said last week that the talks were taking place and the sale is expected to be confirmed on Monday.
But BBC business editor Robert Peston said no UK tax would be paid on the enormous transaction.
He said that as the deal would involve the sale of a stake in a US group, the holding company for which is in the Netherlands, the UK taxman would not be entitled to anything.
This could prove to be controversial, with Vodafone’s tax affairs having been subjected to scrutiny in the past.
Vodafone warned there was still no certainty an agreement would be reached.
In a brief statement in response to what it called media speculation, it said: “Vodafone confirms that it is in advanced discussions with Verizon Communications Inc regarding the disposal of Vodafone’s US group whose principal asset is its 45% interest in Verizon Wireless for $130bn.
“The consideration would substantially comprise a mixture of Verizon common stock and cash.”
If the deal goes through it will be the second most valuable corporate deal, after Vodafone’s own takeover of Mannesmann of Germany in 2000.
An announcement is expected on Monday, despite US markets being closed for Labor Day.
It would bring to an end a long-running saga, with both Vodafone and Verizon trying to take full control of Verizon Wireless over the years, but having been unable to agree a price.
Richard Dunbar, investment director at Scottish Widows, said Verizon Wireless was now in excellent shape, with 100 million customers.
He told the BBC: “It has been worth the wait [for Vodafone shareholders]. Previously it [Verizon Wireless] wasn’t generating a lot of cash and it wasn’t paying dividends to the parent company and the relationship between the managers was a poor one.
“Recently it has been better – and the business has been performing better.”
Robert Peston says that at least half of the proceeds of the deal will be returned to Vodafone shareholders.
It means tens of billions of pounds being paid to UK investors, which he says is being seen by some as a significant injection of cash into the economy, similar to the Bank of England’s quantitative easing programme.
The rest of the money may be used by Vodafone to fund acquisitions in Europe.
Vodafone’s ownership of almost half of Verizon Wireless stems from the $61bn acquisition in 1999 of a US company called AirTouch, which later merged its US mobile phone business with that of Bell Atlantic.
The deal is not expected to have much effect on Verizon Wireless customers, with Vodafone having had little say in the running of the company.