The Sri Lankan cricketers are 256 for seven in their second innings, a lead of 426 after three days of the second Test against South Africa at Kingsmead in Durban.
Kumar Sangakkara hit a superbly-crafted 108, his 28th Test century.
Their lead of 426 is eight runs more than the highest successful run chase in Test history.
South Africa lead the three-match series after winning the first Test at Centurion by an innings and 81 runs.
The hosts can take heart from having chased down 414 against Australia in Perth three seasons ago, only four runs less than the world record 7-418 by the West Indies against Australia in Antigua in 2002/03.