Agarwal, who has majority control of Hindustan Zinc Ltd through Vedanta Ltd, will make the investment via his family trust Volcan Holdings, Volcan said in a statement after the market close on Wednesday.
Anglo American, which has a market value of around 16.75 billion pounds ($20.55 billion), refused to comment. It has also declined comment on reports that it has rebuffed previous approaches for a tie-up with Agarwal's Hindustan Zinc.
Wednesday's announcement of Agarwal's plans said neither Volcan nor Vedanta intended to make an offer for Anglo American.
Anglo's shares were up nearly 10 percent by 1015 GMT, outperforming the broader sector, which rose 6 percent. Vedanta's shares in London also rose around 6 percent.
Last year, Anglo American's shares gained nearly 300 percent, making it the best performer on the London FTSE as the mining industry recovered from a slump in commodity prices in 2015 and early 2016.
In February, it said would resume paying dividends and slow down asset sales as it was no longer under financial pressure.
Analysts are cautiously optimistic about the sector's recovery, although some commodities have stronger supply-demand fundamentals than others.
Zinc is potentially one of the strongest and is also a focus of miner-trader Glencore, which this week increased its control of the market through a deal with Canada's Trevali.
Christopher LaFemina, analyst at Jefferies, said the mining sector, which spent last year putting its balance sheets in order, might be making an early return to M&A activity. Many had not expected that until next year.
"Our expectation has been that the mining sector recovery would comprise three separate phases," LaFemina said, referring to balance sheet recovery, then improved cash flow and finally M&A. "It is possible that Phase 3 is beginning now."
Agarwal, who has four decades of experience as an entrepreneur, is founder and chairman of Vedanta, which has copper operations in Zambia and a zinc mine in South Africa, Anglo American's heartland.
One industry source who has worked with him in the past, speaking on condition of anonymity, said Agarwal had been looking at Anglo American for at least five years.
"His dream is to have Vedanta as one of the big diversified miners at some point," the source said.
A full-scale takeover of Anglo American would be difficult as Vedanta, with a market capital of around 2 billion pounds, is much smaller than Anglo American.
Analysts see Agarwal's move as a new attempt to fulfil his ambition to have a major footing in South Africa as well as in his native India, which he has aired in the Indian press.
"We see this move as potentially forcing either Anglo's, or a rival bidder's, hand," Paul Gait, analyst at Bernstein, wrote.
"From a strategic point of view, some kind of deal could potentially hand Anglo a solution to their problem of perceived 'overweighting' towards South Africa," he said.
Innovating financing
Another industry sources said Agarwal was limiting any risk by using a convertible bond rather than cash to finance the deal. The mandatory exchangeable bond for 2 billion pounds is due in 2020 and is led by J.P. Morgan.
Volcan will issue the bond to fund the share purchase and it will be secured by the purchased shares.
One of the sources said it was an efficient way to buy a sizeable stake as acquiring around 12 percent of a company could be difficult to achieve without attracting attention.
The structure of the bond limits any downside, the sources said, adding the advantage of buying into Anglo American, whose portfolio includes diamonds and platinum, was to diversify Agarwal's holdings.
"This is an attractive investment for our family trust ... I am delighted to become a shareholder in Anglo American plc," Agarwal said in Wednesday's statement.
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