The firm is understood to be in discussions with Littlewoods executives and has tabled an offer of about pounds 450m for the 135-strong high street chain, which was put up for sale last week.
Competitors in the auction are rumoured to include Tesco and possibly another large high street retailer.
A spokesman for Littlewoods said that discussions were taking place with two parties, and the company would be making an announcement in the next two weeks.
This is the second time that KKR has targeted the Liverpool-based pools and stores group. In late 1995 it initially teamed up with former Littlewoods chief executive Barry Dale to mount a pounds 1.2bn takeover bid for the whole company, but later decided to go it alone, before the Moores family, which controls Littlewoods, decided that the company was not for sale after all.
This time analysts believe that the break-up opportunities presented by Littlewoods, with a number of large high street sites including a flagship store on Oxford Street, has once again tempted KKR back to the table.
A spokesman for KKR in London would not be drawn on whether it was in negotiations with Littlewoods. "I am not saying we are in, but I am not saying we are out either," the spokesman said. Littlewoods declined to comment.
Henry Kravis, KKR's chairman, said in November last year that he was keen to spend some of his $6bn war chest on large acquisitions in the UK. If he can pull off the Littlewoods deal, it will be the largest he has executed on this side of the Atlantic, ahead of KKR's pounds 305m purchase of Westminster Press from Pearson.
However, such deals pale in comparison to KKR's $31bn mega buyout of tobacco and food giant RJR Nabisco in 1989, immortalised in the book Barbarians at the Gate.