The Times, citing two sources with knowledge of the deal, said the sale could be announced as soon as Thursday. The deal values Michael Foods at $2.5 billion, one of the sources said.
Shares of Post were up about 7 percent on Wednesday after The Wall Street Journal earlier reported the pending deal.
Representatives of Post and Michael Foods did not respond to requests for comment from several media outlets.
The deal would allow St. Louis-based Post, which is best known for cereals like Raisin Bran and Honey Bunches of Oats, to expand its offerings to include more protein and dairy products.
If it's approved, the deal would nearly double the size of Post, whose market value is about $2 billion, the Wall Street Journal reports.
Michael Foods produces and distributes food products under a variety of brands, including Abbotsford Farms eggs, Simply Potatoes, Crystal Farms cheeses and liquid egg whites. The company reported $1.9 billion of net sales for the year ended Dec. 28, an increase of 5 percent from the year earlier.
Michael employs about 3,500 people, according to a profile in Business Week.
Michael Foods has been under private equity ownership for the past decade, according to the New York Times. It was bought in 2003 by Thomas H. Lee Partners, a Boston-based private equity firm. Lee sold a majority stake to the private equity arm of Goldman Sachs in 2010. It was valued at $1.7 billion in the 2010 transaction.