"There is no plan B. This is a take-it-or-leave-it offer," Oppenheimer said in an interview with the daily.
Oppenheimer is leading DB Investments (DBI), a consortium comprising his wealthy family, Anglo American and the Botswana government. The Oppenheimers, Anglo and Botswana already own 40% of De Beers.
De Beers confirmed on Friday that a shareholder vote on the new offer would take place on May 18 in Johannesburg. Investors had been scheduled to vote on the original offer today.
The consortium, pushed by major investors to raise its initial $17.6bn offer, said last Friday it would raise its cash-and-shares bid by $2 per De Beers share.
On Monday this week, the group said the terms of the revised bid would be 0.446 of an Anglo share plus $15.35 in cash for each de Beers share, compared with 0.43 of a share and $14.40 cash previously.
De Beers shareholders are also to receive a $1 special De Beers dividend and an Anglo final dividend of $1.30 per Anglo share received.
The consortium said the new offer valued De Beers at $45.94 a share.
De Beers shares have been under pressure since Monday on disappointment over the cash component of the revised offer.
The stock ended on Thursday at R320.80, after falling 5% during the session. On Friday at 1230 De Beers shares were trading 1% higher while the JSE all share index was up 0.6%.
BusinessDay quoted Oppenheimer as saying that conditions were such that, as a member of the Oppenheimer family, he was unwilling to top the bid offer by $2 a share.
He said he could also not support a full sale of De Beers' Anglo stake since the offer on the table would also solve the problem of investor-unfriendly cross shareholdings that have depressed shares of both groups. De Beers holds 35.4% in Anglo and Anglo holds 32.3% of De Beers.
Selling the De Beers stake would bring in inflows of $1.1bn into Anglo, Oppenheimer told the daily.
He said the Reserve Bank and the South African Revenue Services were also unlikely to sanction a full sale of the Anglo stake given that the consortium offer would bring in around $3bn of estimated capital inflows.
"We certainly won't see the DB Investments bid voted down and 24 hours later have an unbundling proposal put on the table. This will be a long and drawn-out process to find alternatives," he said.