Owner General Motors wouldn't comment, but the newspaper said the Carlyle and Onex bid was awaiting approval from the United Auto Workers last week.
The other bidders-a party that included Greenbriar Equity, Clayton Dubilier & Rice, and Berkshire Hathaway-pulled out of the auction. That left Carlyle and Onex as the winners, two sources said.
Allison likely will sell for about 10 times the $550 million in projected 2007 earnings before interest, taxes, depreciation and amortization, one of the sources said, although another source put the value at 7.2 times EBITDA.
At 10 times EBITDA, the sale price would come to about $5.5 billion.
The Detroit automaker said in January it would consider strategic options for the maker of truck transmissions as part of an overall push to shed non-core businesses.
The unit is based in Indianapolis, and employs 3,400.
Carlyle is headquartered in Washington, D.C., and Onex, in Toronto.