It would mean the third and fourth largest mobile phone networks respectively combining to create a business with 27 million customers, larger than current leaders BT, EE and Virgin Media O2.
The companies say it would accelerate the rollout of 5G and rural broadband.
Any deal would be scrutinised by the Competition and Markets Authority (CMA).
Reports suggest the two companies are hopeful of striking a deal by the end of the year.
Vodafone said it would own 51% and Hutchison – which operates Three – 49% under the deal being discussed.
Chances of approval?
Regulators have previously opposed mergers that would reduce the number of networks in the UK.
However, Vodafone has pointed to a recent report from communications regulator Ofcom which might suggest a new approach.
It found that both Vodafone and Three had in recent years delivered returns on investments that were lower than the cost of the capital they used.
But James Robinson, from communication market analysts Assembly, said there would still be hurdles to overcome for the merged firm.
“While the parties might view consolidation as a way to improve returns and unlock shareholder value, we expect the CMA would be eager to protect against the risk of consumer price rises – particularly in light of the current cost-of-living crisis,” he said.
He suggested the merged firm may need to provide “legally-binding concessions” to reassure regulators.
‘Motivation is scale’
Ben Wood, an analyst at CCS Insight, told the BBC that the potential tie-up was to be expected.
“The two companies have made no secret of their interest to consolidate,” he said.
“The leading motivation to join forces is scale. In telecommunications, the most successful companies tend to be the largest; bulking up would offer many synergies and cost-saving opportunities.
“Under the status quo, it’s hard to see either operator growing enough organically to get close to challenging BT and Virgin Media O2 for size in the UK.
Vodafone pairing with Three is just one potential tie-up in the UK.
Other deals involving Virgin Media O2, TalkTalk and Sky have been speculated in recent months.
“By combining our businesses, Vodafone UK and Three UK will gain the necessary scale to be able to accelerate the rollout of full 5G in the UK, and expand broadband connectivity to rural communities and small businesses,” Vodafone said in an update to shareholders.
“The conditions to ensure thriving competition in the market need to be nurtured, otherwise the UK is at risk of losing the opportunity to be a 5G leader,” it added.
Ofcom said it did not have a fixed position in relation to market consolidation.
“Our view is that potential mergers in telecoms markets need to be assessed on a case-by-case basis, rather than on a presumptive view of the appropriate number of competitors,” it said in its latest report.