Citing people familiar with the matter, the Journal said Nestle appears to have outstripped Danone
Nestle, the world's biggest packaged food company, was already seen as a favoured bidder due largely to its deep pockets.
Bloomberg reported that Danone and Nestle both submitted final offers of about $10 billion for the business, citing people with knowledge of the matter.
According to that report, Danone has told outsiders that it is unlikely to win the unit, but that Pfizer is trying to keep them in the auction, one of the people said. Bloomberg said a third bidder is no longer in the running.
Separately, Pfizer has hired JPMorgan Chase & Co
Pfizer put its infant nutrition and animal health businesses up for sale last July as it looks to focus on its core pharmaceuticals business.
A company official said on Tuesday that no decisions have been made and that Pfizer will be in a position to announce any decisions this year. The company expects to complete any transactions between July of this year and July 2013.
Regarding the animal health business, the official said a spin-off was most likely.
A Nestle spokesman was not immediately available for comment.
A person familiar with the negotiations said a deal for the infant nutrition unit was expected in the near future, but not this week.
Pfizer's infant nutrition business sells baby formula and maternal supplements. It generates over 70 percent of its $2.1 billion in sales in emerging markets, with over a quarter coming from China, where the $6 billion market is expected to double by 2016.
The Pfizer business competes with products from Nestle and Danone, as well as Mead Johnson's Enfamil and Abbott Laboratories'
Pfizer shares closed 1.5 percent higher on Tuesday.
(Reporting by Olivia Oran, Ransdell Pierson, Martinne Geller and Michael Erman in New York; Editing by Gunna Dickson, Bernard Orr, Tim Dobbyn and Sugita Katyal)