The story, which appeared in the New York Post, did not say whether Kraft Heinz had a hired a bank to explore such interest, a step that often precedes serious attempts at dealmaking.
Campbell previously announced a strategic review in which it would leave "no sacred cows," raising speculation the company could do a transformative deal. The soup company has long been a takeover target, but its largest shareholders, the Dorrance family, has not previously indicated a willingness to sell.
Still, pressure on Campbell is immense. Its efforts to move beyond packaged soup, by going into businesses like fresh food, have been challenged operationally. Efforts to augment its faster-growing snack business through a roughly $6 billion acquisition of snack company Snyder's-Lance, left it with more debt and a business that sources tell CNBC will be difficult to integrate.
Campbell CFO Anthony DiSilvestro recently told analysts during an earnings conference call that the work performed since the deal closed has confirmed the company's initial synergy assumptions.
So far, Campbell has kept all strategic options on the table and it has yet to formally hire a bank, sources familiar with the situation said. It plans to have a board meeting this week, sources tell CNBC's David Faber.
A Campbell spokesman said that he "would not speculate on the outcome of [Campbell's] previously announced strategic review."
The soup company has said it plans to announce results of its review at the end of August.
One idea that has frequently been pitched is the soup company splitting up its faster-growing snack unit and its slower-growing soup business, sources said. Such a move would follow a similar path paved by Kraft when it spun out its Oreo snacking business, Mondelez, in 2012. Some have also suggested the soup business could then go private by partnering with a private equity firm, thereby allowing it to make significant strategic changes out of the spotlight of the public eye.
While the slow growth of the soup business has weighed on its stock price, it remains profitable, a desirable characteristic for private equity firms.
Meantime, while Kraft Heinz has long been pointed to as a potential buyer of Campbell, should it ever be for sale, several industry sources have questioned whether Kraft Heinz would want to make a bet on a slow-growing company, as it grapples with its own growth challenges. Kraft Heinz's backers, 3G Capital, are adept at cutting costs, but have not yet proven themselves in their ability to grow brands, a challenge that has plagued the food industry at large.
As such, their dealmaking focus has included other deals beyond the mega-deals for which they have become known. As an example, Kraft recently looked seriously at acquiring yogurt company Noosa Yoghurt, CNBC has reported.
And while the firm is known to be a serial acquirer, it is also known to be thoughtful in its dealmaking.
Meantime, 3G's other prominent characteristic, a reputation for cost-cutting, may make the Dorrance family resistant to sell their business to the ketchup maker, industry sources said.
Even with Monday's gains, Campbell shares are down 21 percent in the past year, giving the soup maker a market capitalization of $12.7 billion. That compares to market caps of $77.2 billion for Kraft and $61.1 billion for Mondelez. Campbell shares peaked at $54.37 last August.
— CNBC's David Faber contributed to this story