The implied offer price is R347 per De Beers share but some of De Beers' biggest shareholders want a higher return and would prefer to see a straight unbundling of the unpopular cross-holdings between De Beers and Anglo, with De Beers emerging as a stand-alone listed company, even though this could result in a hefty tax bill and regulatory hurdles.
A spokeswoman for DB Investments - the Oppenheimer family-led consortium orchestrating the buy-out - confirmed on Tuesday that Oppenheimer had begun talks in Cape Town on Monday with shareholders opposed to the deal. The talks will continue until Wednesday.
As yet there are no details as to Oppenheimer's progress with the defiant shareholders.
A Johannesburg-based analyst who works for one of the companies that hold shares in De Beers said institutional investors felt the $17.6bn offered by Anglo and the Oppenheimer family was too low.
These investors argue that the shares have historically been undervalued and that the Oppenheimer family would not be trying so hard to buy them back if they were not worth considerably more.
"There are those who felt that regardless of any offer made, it might still not be enough to change their minds to vote in favour of the proposed deal," said the analyst, who did not want to be named.
Some shareholders also felt that the proposed buy-out did not constitute shareholder democracy by Anglo, which owns a 45% stake in De Beers.
The deal would unravel complex cross-holding between De Beers and Anglo American which have been held for the past 75 years.
If it goes ahead, De Beers - the cash cow in the diamond and gold empire - will become a private company run by the Oppenheimers, the South African dynasty which holds controlling stakes in both mineral giants.
In return, the family would relinquish part of its stake in Anglo American, the company it founded in 1917.
The offer received a double blow last week when two big US shareholders said they could not accept the Oppenheimer offer.
This was followed by reports from Cape Town that shareholders were banding together to force Anglo American to unbundle its shareholding in De Beers rather than taking private control of the company.
Shareholders have long felt that De Beers' shares might be worth more on diamonds alone if they were not tied to the performance of the rest of the Oppenheimer empire with its interests in gold and platinum.
Anglo American is not allowed to vote at the May 4 meeting in Johannesburg where a 75% vote in favour of the deal is needed for it to go ahead.
De Beers shares were down R4 at R317 on Tuesday following a drop of 1.5%, or R5, on Monday.
Anglo shares edged up 0.7% on Tuesday to close at R515 after ending unchanged on Monday. -Sapa-AFP