TWE, which owns brands such as Wolf Blass, Rosemount and Lindeman's, is the world's biggest listed winemaker.
Last year, it sold 385 million bottles and recorded revenues of AU$1.76bn.
KKR's new bid is a joint venture with fellow US private equity firm Rhone Group.
Rhone Group's contribution is not known and the bid is being led by KKR.
The proposed offer is worth AU$5.20 a share and represents a 11% increase on the earlier April bid, which was rejected by TWE's board.
KKR had a large cash influx following the $5.1bn (Â£3bn) sale of Oriental Brewery of South Korea to Anheuser-Busch InBev and appears thirsty for another drinks company.
The winemaker's board has urged shareholders not to throw out the new proposal.
It said in a statement: "The board of TWE, together with its advisers, has concluded, based on the revised proposal, that it is in the interests of its shareholders to engage further with KKR and Rhone."
TWE has a turbulent history and is currently undergoing restructuring that will see 175 job cuts.
It formed in 1995 as the wine division of Australia's Foster's group, brewers of the eponymous beer. However, continuing losses led to it being demerged in 2011.
Recent results have been disappointing, with volume falls in Australia, the result of risky price strategy and underwhelming sales in the China the result of government austerity.
In June last year, the group was forced to destroy AU$34m worth of unsellable wine as a result of disappointing US sales.

The changes will see the workforce drop by 85,000, from 135,000 to 50,000.
The group has been forced to restructure after the collapse of commodity prices slashed profits.
Anglo will also suspend dividend payments for a year, and consolidate from six to three businesses.
All the world's big mining companies have seen profits tumble along with plunging commodity prices as demand from China has slowed.
The price of oil is at seven-year lows, as is copper, and on Tuesday the price of iron ore tumbled to a 10-year low of $39.60 a tonne, after reaching a peak near $200 in 2011.
As part of the restructuring, Anglo American's diamonds business will be run by its De Beers subsidiary, its platinum and base metals operations will come under Industrial Metals, and its Bulk Commodities division will concentrate on coal and iron ore.
A company spokesperson said the job cuts would be made through asset sales and internal cuts:  "Bear in mind that these include assets that we will sell, so the 85,000 jobs don't [all] disappear as many will be employed by new owners of  those mines that we sell."
Anglo will sell its phosphates and niobium businesses during 2016 and plans $3.7bn in cost and productivity improvements by 2017.
Chief executive Mark Cutifani said: "While we have continued to deliver our business restructuring and performance objectives across the board, the severity of commodity price deterioration requires bolder action."
Anglo's share price has fallen by more than two-thirds over the past year. Today they fell 12.3% to 323.65 pence. In February 2011, the shares traded for more than Â£34 each.
Shares in other major mining companies also saw sharp falls in trading in London on Tuesday. Glencore was down 9.3%, Rio Tinto fell 6.3% and BHP Billiton was 5.8% lower.
"Anglo American, like all of its counterparties in the mining sector, has found there is no more fat to be cut and, as had been speculated for the last week, it had to face the reality that it could no longer pay out the dividend," said Alastair McCaig, market analyst at IG.
"Where one goes, others will follow and the possibility that BHP Billiton or Rio Tinto might be forced into similar action now looks increasingly likely."
US private equity giant KKR has launched a renewed $3.4bn Australian dollar (Â£1.88bn; $3.17bn) takeover bid for Australian winemaker Treasury Wines Estates.
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Shares in mining firm Anglo American have fallen to a record low as the company said it would sell huge chunks of its business and shrink its workforce by nearly two-thirds.