When Miami Beach police officer Ysidro Llamoca encountered Jamie McNeil — shirtless, shoeless, unshaven and scruffy — the beach dweller was preparing to sleep again on the sand dunes of Lummus Park.

Instead of hassling the homeless man, Llamoca offered to help. Fifteen minutes later, McNeil, 34, was in a white van operated by the city’s homeless outreach team, headed for shelter at the Homeless Assistance Center in Miami.

The treatment last week was a far cry from the claim that Miami Beach officers ferry their homeless people across the MacArthur Causeway and deposit them on downtown Miami streets. That assertion has been repeated by Miami Commission Chairman Marc Sarnoff as part of his recent quest to banish homeless people from the struggling downtown business district.

“We’ve caught Miami Beach dumping people in the city [of Miami],” Sarnoff recently told the Miami Herald, accusing Aventura, Surfside and Coral Gables police of the same practice. “Downtown Miami has become the dumping ground for Miami-Dade’s homeless.”

But the cities he cited deny the claim. And while advocates for the homeless have long heard rumors about the practice, they say there is no evidence to support it.

“I’ve never seen it or received any actual reports of it happening,” said Hilda Fernandez, executive director of the county’s Homeless Trust.

The more likely reason that so many homeless people congregate in Miami, Fernandez said, is that Jackson Memorial Hospital, the county courthouse and several crisis intervention centers are all located downtown. Downtown Miami also has a Greyhound station. The modestly priced bus service draws homeless people traveling to and from Miami.

The Homeless Trust is scheduled to meet Monday to discuss, among other things, a proposal to fund more shelter beds in downtown and to finalize its budget.

Claims of homeless dumping aren’t new. In arguing the need to ban street feedings in 2004, former city manager Joe Arriola complained that Miami had long been a dumping ground for the region’s homeless.

The debate found new life in April, when the city commission decided to petition a judge for changes to a landmark 1998 legal settlement known as Pottinger v. Miami. The agreement prohibits Miami police for arresting homeless people for minor offenses without first offering them a bed in homeless shelter.

Sarnoff, who also chairs the Downtown Development Authority, argues that the strict parameters of Pottinger have prevented city leaders from removing the 500 or so homeless people from Miami’s streets. Most local shelters are at capacity, meaning police can do little to punish the homeless who urinate in the street or light cooking fires in public parks.

Sarnoff pointed out that other municipalities aren’t subject to the same rules, and said they sometimes resort to escorting their homeless residents to Miami. That claim, he said, is based on anecdotes he has head from Miami police officers and several of his downtown constituents.

Miami Police spokesman Napier Velazquez said the department has “no documentation” of other municipalities releasing homeless people on the streets of Miami. The anecdotes, however, include a first-person account from attorney Jay Solowsky, who said he saw a Coral Gables police officer drop off a homeless person near 150 W. Flagler St. late one night in 2009.