Authorities including the National Transportation Safety Board are investigating Thursday's crash.
There has been no word of a cause.
NTSB investigators were examining the remnants of the helicopter on Friday and planned to brief reporters.
The victims included Siemens executive Agustin Escobar, his wife Mercè Camprubí Montal, who was a global manager at an energy technology company, and their three children, in addition to the pilot.
New York City Mayor Eric Adams said the children were four, eight and 10 years old, and the eight-year-old's birthday was Friday.
"It's just a real unfortunate situation. And our heart goes out to the family members," Adams told Fox 5 New York.
Escobar was in the New York area on business and his family flew over to meet him for a few days, said Steven Fulop, mayor of Jersey City, New Jersey, in a post on X.
He said a relative would arrive on Friday and that officials were working with the medical examiner to release the bodies for their return to Spain.
Pieces of the aircraft could still be seen floating in the river on Friday morning as divers resumed the search for evidence.
The bulk of the cabin was retrieved by crane on Thursday night.
Community activists and officials have repeatedly proposed banning or restricting traffic at Manhattan heliports, citing New York City's history of fatal helicopter accidents that have killed 38 people since 1977 as well as the relentless noise.
In an interview with the Associated Press, state Senator Brad Hoylman-Sigal said the crash was a "reminder of our worst fears of tourist helicopters" and called for restrictions on flights over the city.
"Having non-essential flights over densely populated areas is a recipe for disaster," he said.
Robert Carroll, a Democratic member of the state Assembly, called for a stop to all sightseeing and commuter-related helicopter flights until the cause of Thursday's crash is determined.
Adams, however, said he does not want to restrict such flights until the investigation is complete.
"We have thousands of flights using helicopters and tourism," he said on 1010 WINS news radio.
"People want to see the city from the sky. At the heart of this is safety. It must be done right. We must look at the maintenance record of this helicopter. We must look at the record of the pilot. And we need to find out what happened."
A helicopter industry group also urged caution.
"The helicopter community is in shock and mourning after the tragic and horrific events of yesterday," Jeff Smith, chairman of the Eastern Region Helicopter Council, said in a statement.
"Unfortunately, some well-meaning but misguided leaders are using this tragedy to exploit and push their decades-old agenda to ban all helicopters".
