The Zaandam, operated by Holland America Line, and its sister ship the Rotterdam are expected to dock in Fort Lauderdale, after days of protracted negotiations over their fate.
A commissioner of Broward County, which includes Fort Lauderdale, said on Twitter that the ships had received "conditional approval" from local authorities, the Coast Guard and health officials to come into port.
"Look forward to seeing a SAFE plan for all to resolve," said the commissioner, Michael Udine.
Holland America, which is owned by Carnival, said the ships would "remain outside US waters while awaiting clearance to enter".
Florida's Republican governor, Ron DeSantis, initially said he did not want the ships to dock, for fear the ill passengers would tax the state's already strained health care system.
With more than 7,700 coronavirus cases and more than 100 deaths, the Sunshine State -- home to many retirees -- has the fifth-most cases in the United States.
But late Wednesday, DeSantis told Fox News that he had not realized there were US citizens involved.
"We actually have Floridians" aboard the Zaandam, he said.
President Donald Trump also said the ships needed to be evacuated, saying: "We have to help the people. They're in big trouble."
Trump said he was working with British and Canadian authorities to repatriate their nationals who are on the cruise liners.
How to disembark
A total of 1,243 passengers and 1,247 crew members are stranded at sea on the Zaandam and the Rotterdam, which came to its sister ship's aid last week, loaded with supplies.
About 1,200 passengers who are not ill would be sent home on charter planes. They will be "transported in coaches that will be sanitised, with limited person-to-person contact and while wearing masks", Holland America said Wednesday.
Some 45 people with mild symptoms will remain aboard in isolation until they recover and the estimated less than 10 people requiring critical care will be taken ashore for treatment locally, the company said.
The Sun-Sentinel, one of south Florida's biggest newspapers, urged Carnival to come up with a better plan for keeping passengers and Floridians safe.
"A better idea would be to quarantine the seemingly healthy passengers for the recommended 14 days," it said late Wednesday in an editorial.
Cursed ship
The desperate situation aboard the Zaandam attracted worldwide publicity, but it is just one of several cruise liners seeking permission to dock at Port Everglades in Fort Lauderdale.
The Zaandam, which left Buenos Aires on March 7, was originally meant to finish the trip in Chile on March 21, but changed course due to the virus.
After negotiations while the ships waited in waters off Panama City, they were allowed to transit the Panama Canal in order to head to Florida.
It was originally scheduled to arrive in Fort Lauderdale on April 7 but that timetable was revised as the situation on board worsened.
Four people have died on the ship, for reasons not yet disclosed.
Rick De Pinho, a 53-year-old attorney, and his wife were transferred from the Zaandam to the Rotterdam because so far, they are healthy.
"You can't have these ships floating around. People want to go home," he told AFP from the ship.
His wife Wendy, 49, works for a financial services company.
She urged politicians to make the "right humanitarian and financial decisions," noting how much revenue the cruise industry generates in Florida.
"Please let us dock," she said.