BOSTON — A proposed skyscraper is threatening to throw shade on two of America’s most treasured parks, Boston Common and the neighboring Public Garden, in exchange for a windfall payday for the city.

At issue is whether developers of a proposed $1 billion, 775-foot tower — what would become Boston’s third-tallest building — should be allowed to violate a state law banning buildings from casting shadows over historic parks.

“Hang out and watch people. People seek the sun. If a person is sitting on a bench and a building casts a shadow, they’ll move down the bench into the sun,” said Elizabeth Vizza, executive director of Friends of the Public Garden. “They always chase the sun.”

The 50-acre Boston Common was founded in 1634, making it the country’s oldest city park. It has hosted everything from grazing cows and public hangings to concerts and protests.

The 24-acre Public Garden may be even more treasured, with its human-pedaled swan boats, many tree species and duck-size bronze statues based on Robert McCloskey’s children’s classic, “Make Way for Ducklings.”

City officials say the $153 million that developer Millennium Partners has offered for the city-owned parcel on which the tower would stand would go to parks and affordable housing. The parcel now is home to a crumbling garage several blocks from the common.

For backers, the longer shadows seem a small price to pour millions back into the neighborhoods.

Mayor Martin Walsh already committed $67 million of the money to the Boston Common, Franklin Park — a sprawling 527-acre park in the Jamaica Plain, Roxbury and Dorchester neighborhoods — and the Emerald Necklace, a string of green spaces designed by the famous landscape architect Frederick Law Olmsted.

Another $35 million will be dedicated to upgrading two public housing developments in the South Boston and East Boston neighborhoods.

The proposal has already been approved by the city council but still must win approval from state lawmakers and Republican Gov. Charlie Baker.

Vizza said the deal effectively puts a price on the cost of shadows for future developers looking for exemptions. More sunlight isn’t just good for the trees, she said, but for psyche of thousands of people who visit the parks every day.

The exemption will violate the shadow law 264 days of the year on the Boston Common, and 120 days on the Public Garden, Vizza said.

Millennium Partners said its simulation shows the longest the tower’s shadow would cover portions of the Boston Common would be for about 90 minutes in the morning in early September. The longest the shadow would cover parts of the Public Garden would be in the morning in late August for about 26 minutes.

Brian Golden, director of the Boston Planning and Development Agency, said the tower poses a unique opportunity because it’s being proposed on a publicly owned parcel in the middle of a dizzying building boom and will replace a dilapidated, condemned structure — all at the cost of what he describes as fleeting morning shadows.

“These are sacred spaces in many ways. We care just as much about protecting those two parks,” he said. “But a fairly insignificant shadow does not harm them in any way, but does unlock the potential of tens of millions for needs across the city.”

Alice Galakatos, a Beacon Hill resident who exercises her 6-year-old Jack Russell terrier, Roxy, on the common, said she’d prefer sun over shadows.

“I’d rather have sun. I’d rather have sun all the time. Who doesn’t like the sun?” said Galakatos, 54.

But Galakatos said she can also see the city’s point of view.

“I’m just thinking of it from my perspective, my point of view coming out here with the dog, but it’s just a little tiny perspective compared to what it brings in” for the city, she added. “I could see how that could balance out. It’s a trade-off.”