The mine is estimated to be worth between $750m and $1.2bn, it said.
The sale had also attracted initial interest from private equity firm Apollo Global Management, the newspaper reported.
In April it was reported that KKR was looking to buy the diamond operations of BHP Billiton and Rio Tinto and merge them to capitalise on the recent surge in demand for jewels.
BHP put the unit on the block last year in a bid to focus on its core commodity mining operations such as iron ore and oil.
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