In a candid moment, the Jolly Green Giant might be saying “ho-ho-hum” these days.

Frozen and canned vegetable sales are falling. Green Giant’s strong suit — frozen veggies packed in sauce — has been particularly hard hit. There’s even speculation that General Mills Inc., Green Giant’s owner, might sell the venerable brand with deep Minnesota roots.

Golden Valley-based General Mills won’t comment on sale speculation. But with a major product launch due this spring, the company hasn’t given up on Green Giant.

“Vegetables are a great place to be right now,” said Ray Joncas, marketing director for Green Giant. “They are healthy and people are eating more of them.”

As consumers increasingly opt for simpler, less processed foods, vegetables are indeed a bright spot in the food business. But the star of the category is fresh produce.

Last year, fresh vegetable sales grew 4.4 percent, while frozen and canned sales respectively declined 2.5 percent and 1 percent, according to market researcher Nielsen. Over the past four years, sales of frozen veggies have on average been flat.

“Produce is a department that has experienced a lot of growth in the last five years,” said Jonna Parker, director of Nielsen Perishables Group. There are more varieties — kale anyone? — and more mass market retailers are enlarging their fresh vegetable sections.

Trends in the vegetable market mirror shifts throughout the food industry. The grocery store’s periphery — home to fresh produce, meat and dairy products — has been gaining sales. The “center store” — home to more processed foods — has been losing its luster.

General Mills has a foothold in fresh vegetables, but it licenses the Green Giant brand to independent produce companies. Frozen vegetables, which Mills packages itself, are the anchor of the Green Giant business, and canned product is still important. Minnesota is the primary source of Green Giant canned corn and peas.

The Minnesota Valley Canning Co. was founded in 1903 to capitalize on the state’s vegetable bounty. It created the Green Giant brand in the 1920s. The company renamed itself Green Giant in 1950 and was sold to Minneapolis-based Pillsbury in 1979, which in turn was scooped up by General Mills in 2001.

Green Giant does $600 million to $700 million in annual sales. Its stronghold is prepared or “sauced” vegetables — frozen cauliflower with cheese, creamed spinach and so on. Sold under the Green Giant Steamers label, sauced offerings aren’t commodity products like plain frozen vegetables, and therefore tend to carry higher profit margins.

Green Giant has a commanding lead in sauced vegetables with a 76 percent market share for the year ended Feb. 22, according to researcher IRI, which tracks sales at food retailers. The Birdseye brand, which leads frozen vegetables generally, is a distant second in sauced with a 17 percent share.

But sauced vegetables have lost their zest: Sales for the category fell 15 percent during the year, according to IRI. To make matters worse, General Mills’ prices for a key product were out of whack to competitors.

“What is frankly declining is heavily sauced vegetables, and that is where Green Giant has historically played and done well,” said General Mills’ Joncas. “Heavily sauced is declining because consumers want things that are simpler and less processed.”

General Mills fell behind in frozen vegetable innovation. “There are things consumers want from the frozen category, and we haven’t been fast enough to give them what they want,” Joncas said.

General Mills is trying to do something about that. Since last fall, the company has been testing pre-roasted (and unsauced) Green Giant vegetables in the Southeast. In May, General Mills will introduce a new line called “Sautés by Green Giant,” five frozen medleys that instead of sauce feature packets of dry seasonings and garnishes.

Like General Mills’ new pre-roasted vegetables, the Sautés are meant to be stove-cooked in a pan, not in a microwave oven. They’re also meant to be “restaurant inspired” with the type of “bold seasonings” that are increasingly popular in packaged food, Joncas said.

The Sautés lineup includes Bacon Brussels Sprouts and Indian Spiced Lentils, the latter a combination of carrots, cauliflower, spinach, onions, lentils, cranberries and an almond garnish.

The product plays into broad ingredient trends in packaged food, from ancient grains bread to sriracha-flavored everything. “I think you are seeing packaged food companies be more innovative from a flavor and taste perspective,” said Erin Lash, a stock analyst at Morningstar Inc.

While moving ahead with new products, General Mills still might be prepared to unload its vegetable business. Reuters, citing unnamed sources, reported earlier this month that General Mills has retained an investment bank to peddle Green Giant.

The sale of Green Giant wouldn’t be surprising given General Mills’ struggles in frozen vegetables, Christopher Growe, a stock analyst at Stifel Nicolaus wrote earlier this month. He estimates Green Giant makes $125 million in annual earnings before interest, taxes, depreciation and amortization. At that profit level, Green Giant could fetch $1.3 billion to $1.5 billion, Growe wrote.

Dealmaking activity is heating up in the packaged food industry. Last week, H.J. Heinz and Kraft Foods announced a megamerger worth $46 billion. The industry is twittering with rumors about who’s next, who’s buying and who’s selling.

But if Green Giant is for sale, General Mills might find only a limited number of potential buyers, depressing potential bids, said Jack Russo, an analyst at Edward Jones. “When you go to sell something and it’s not in vogue or out of favor, what kind of price are you going to get?”