Kraft Heinz, the ailing food giant, has tapped investment bank Credit Suisse to review options for its Maxwell House coffee business, which could include a potential sale, people familiar with the matter tell CNBC.

The coffee business has roughly $400 million in earnings before interest, taxes, depreciation and amortization, the people said.

Based off valuations for other sales of consumer brands, a sale could fetch a price of at least $3 billion, they said, though cautioning its price would depend on buyer interest. The coffee industry has become more challenging in the past few years, but private equity firms have shown interest in buying some large, tired brands. Private equity firm KKR last year paid roughly $8 billion for Unilever's I Can't Believe It's Not Butter and other spreads business.

The sale of the coffee business will be one of a string of divestitures for Kraft Heinz, the people said, as it looks to reshape the empire put together by its private equity backer 3G Capital.

3G Capital, which, along with Berkshire Hathaway, bought H.J. Heinz in 2013 and merged it with Kraft two years later, has made its name in splashy acquisitions of U.S. consumer companies. It put together Restaurant Brands International, owner of Burger King, Tim Hortons and Popeyes and created beer giant Anheuser-Busch Inbev. But trends have changed since 3G Capital first pounced on the U.S. scene. Bigger is not better, as large brands fend off competition from leaner, innovative rivals.

It is a struggle to ignite growth when a company is saddled with large, off-trend labels like Oscar Mayer and Velveeta, while facing rising costs. Those challenges have been exacerbated by 3G Capital's cut-to-the bone approach to costs, which some analysts say has come at the sacrifice of its brands' health. The firm extracted $1.7 billion in savings from combining Kraft and Heinz.

Kraft Heinz last week delivered fourth-quarter earnings and revenue that were sharply lower than estimates. Even worse, it slashed its dividend by 36 percent and took a $15 billion write-down on two of its biggest brands, Kraft and Oscar Mayer — an acknowledgement the brands entice shoppers far less than they used to.

As it delivered the bad news, Kraft Heinz executives told investors last week to expect more divestitures going forward to help wipe debt off its balance sheet. The company is aiming to get its leverage down to three times EBITDA, rather than the four times analysts say it is currently pegged at. One notch lower of leverage can be impactful for a company facing margin pressure and profit declines. It could imply asset sales in the billions.

Kraft Heinz last year sold it Canadian dairy business and its Indian beverage business Complan. It said last week it is looking selling brands "with no clear path to competitive advantage."