NEW YORK — If soda-and-snack-food giant PepsiCo were to eat up Oreo cookie-maker Mondelez, as an activist investor is proposing, the deal would reunite two of the most powerful women in the food industry.

Nelson Peltz, an investor who has stakes in both companies, said during an interview on CNBC Tuesday that he wants PepsiCo Inc. to spin off its underperforming beverage business and beef up its Frito-Lay snack business with Mondelez.

The New York native knows both women, having played a behind-the-scenes role in the creation of Mondelez and engaging in talks with PepsiCo about its restructuring options. Peltz has disclosed stakes worth $1.23 billion in Mondelez and $951.8 million in PepsiCo.

Some industry analysts believe a merger is possible.

Consumer Edge Research CEO Bill Pecoriello has noted that the global snack business is much more attractive to PepsiCo than the soda business, which has been declining for years in developed markets such as the U.S.

In fact, Consumer Edge issued a report in March detailing how a merged PepsiCo and Mondelez would work in markets around the world. The report found that the combined company would benefit because there are many regions where one of the companies has a big presence and the other doesn’t.

Little is known publicly about the relationship between PepsiCo’s glamorous CEO, Indra Nooyi, and Irene Rosenfeld, the top executive at Mondelez, who tends to stay out of the spotlight. But the paths they took to the top of their respective companies intersected briefly when Rosenfeld left Kraft to join PepsiCo as head of its Frito-Lay division in 2004.

At the time, Nooyi was serving as PepsiCo’s chief financial officer and was regarded as a strong contender for the top spot. In 2006, Rosenfeld ended up leaving to return to Kraft as CEO; Nooyi was promoted to CEO at PepsiCo less than two months later.

Without providing details, a spokesman said Nooyi keeps in touch with Rosenfeld, with whom she has a warm relationship.

As the chief executives of two of the country’s biggest snack food makers, they’ve dealt with many of same pressures, including the push to make their products healthier and the spotlight that comes with being among the ranks of women CEOs in the Fortune 500.

Nooyi is a high-profile figure whose name has been bandied about for various high-profile positions including the head of the World Bank. The 57-year-old appears at home in front of the cameras, whether she’s making appearances on CNBC’s “Mad Money” with Jim Cramer or talking about her days in a rock band while growing up in India.

She has said that her greatest extravagance is clothes, given the fact that she can’t be seen in public wearing the same outfit too many times.

Rosenfeld, meanwhile, is known among reporters for closely sticking to sound bites in interviews. She doesn’t seem as comfortable around the press, but people who’ve known her say that she is warmer in private.

And although impeccably dressed, the Chinese publication Modern Weekly once noted that Rosenfeld was wearing a brooch she had been spotted wearing in several other published photos.

“You can clearly see where she chooses to devote her time,” the publication wrote in apparent praise.

Even though both women were influenced by music and sports in their youths, the specifics of their experiences diverge. Nooyi played in a rock band and an all-girls cricket team. Rosenfeld still loves playing the piano, opting for John Denver songs but preferring Bach and Vivaldi when she’s listening. In high school, she was the point guard on the basketball team.

In an interview with the Financial Times last summer, she explained why she liked the position.

“You call the shots, you call the plays,” she said. “I kind of went in and out of people’s legs, and I still do that, to a large extent.”

Both are also familiar with the heat that comes when failing to satisfy investors. In late 2011, Nooyi was facing intense criticism because Pepsi had been losing market share to Coca-Cola over the years.

The New York Post — which is known for cartoonish Photoshop creations of top executives — published an image of Nooyi trying to balance on a soda bottle as if it were a surfboard. The nose of the bottle was pointed downward, an apparent illustration of the company’s falling stock price at the time.

A few months later, Nooyi announced the “reset” for PepsiCo that would include a major cost-cutting program and increased investment in marketing for its flagship brands.

While at Kraft, Rosenfeld’s takeover of British confectioner Cadbury to expand overseas was driven in great part by Peltz. The move set the stage for the split of the company last year, with Rosenfeld and Mondelez walking off with global snack brands such as Oreo and Trident.

Kraft held onto slower-growing North American grocery brands such as Jell-O and Miracle Whip.

It’s not clear whether Rosenfeld would stay in her position even if an acquisition were to happen. But under her contract, she’s guaranteed three times her base salary plus her target annual incentive pay if she loses her position within two years as a result of a “change in control” of the company.