The company attracted huge buyout interest last month according to Reuters, which said Advent International, CVC Capital Partners, Clayton Dubilier & Rice and Ares Management had all made it through the first round of bidding.
But it said their interest had now cooled, leaving Bain in pole position to pick up the business through a deal which could be valued at up to $2bn.
Final bids for the business are said to be due this week.
Madison bought into the business in 2006 through a $1.7bn take-private deal, which included the assumption of about $300m in debt.
A $2bn valuation would mean Yankee was worth 10 times its $200m EBITDA, people familiar with the matter previously told Reuters.
The 44-year-old business had a net income of $56.3m in fiscal 2012, up slightly from the $54.5m it registered the previous financial year.
The firm had originally targeted $10bn for its sixth buyout fund, which was launched in 2007 before the financial crisis took hold.
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