The 9/11 Memorial & Museum's funding and leadership could change as the Trump administration explores how it could take over the site in New York City.

A White House official confirmed reports that President Trump's team is discussing ways to shift the nonprofit September 11 memorial in Lower Manhattan from being under New York state control to the federal government.

Gov. Kathy Hochul slammed the president's interest in controlling the site dedicated to those lost in the 2001 terror attacks in a statement issued by her office. 

"Just months after slashing funding for the World Trade Center Health Program that cares for survivors and first responders, President Trump now wants to take over the 9/11 Memorial Museum. The 9/11 Memorial belongs to New Yorkers — the families, survivors, and first responders who have carried this legacy for more than two decades and ensured we never forget. Before he meddles with this sacred site, the President should start by honoring survivors and supporting the families of victims," Hochul said.

It is unclear if Mr. Trump's intentions have anything to do with prior complaints of high ticket prices, executive salaries and the handling of unidentified remains.  

The president previously floated the idea of giving the site national monument status.

9/11 Memorial & Museum leaders don't see purpose for federal takeover 

It is also unclear how a path to a federal takeover of the 9/11 Memorial would be possible. Museum leaders said they know of no legal way to do so. 

"At a time when the federal government is working to cut costs, assuming the full operating expenses for the site makes no sense," Beth Hillman, President and CEO of the 9/11 Memorial & Museum, said in a statement.

Anthoula Katsimatides is a trustee for the memorial and museum. Her brother, John, was a financial trader working for Cantor Fitzgerald when he was killed in the Sept. 11 attacks.

"I would prefer for it to stay as is," she said. "It was our intent to memorialize my brother and all the victims that died that day. It was super important to make sure that it was a hopeful and inspiring place ... and I think it's done in an exceptionally efficient and effective manner."

Brett Eagleson's father was killed in the attacks. His remains were never found.

"It should be left to the 9/11 community. It should be the families' decision," Eagleson said.

Eagleson is part of a coalition of 9/11 families who recently got the green light from a federal judge to move forward with a civil lawsuit against Saudi Arabia for what they say was its role in the September 11 attacks. He says the conversation should focus more on accountability, rather than who regulates the site.

"That should be the dominating story and anything else is a distraction, quite frankly," he said.

The White House did not respond to CBS News New York's request for comment.