Richard Azzopardi, a spokesman for Mr. Cuomo, said the Rosensteins are “longtime friends of Sandra and the governor, with no business before the state.”

Image Barry Rosenstein and his wife, Lizanne, have given Mr. Cuomo a combined $186,000 for his three campaigns for governor. Credit... Rick Wilking/Reuters

“The value of the trip is being tabulated, and as it was not in his official capacity nor related to the campaign, will not be covered by taxpayers nor by campaign funds,” Mr. Azzopardi said. Ms. Rosenstein joined the governor for the trip, he said.

The Rosensteins declined to comment.

Mr. Cuomo and Ms. Lee, a cookbook author who also has a show on the Food Network, have been together for more than a decade. The film, “RX Early Detection: A Cancer Journey With Sandra Lee” is a short documentary about her 2015 battle with breast cancer.

Mr. Cuomo does not frequently travel outside New York but he has occasionally used the private planes of donors. Last September, Mr. Cuomo led a delegation to the U.S. Virgin Islands following a hurricane there and used the jet of John Catsimatidis, a billionaire who has donated more than $130,000 to Mr. Cuomo.

The trip came shortly after Mr. Cuomo had signed biofuels legislation that could benefit one of Mr. Catsimatidis’ companies. Mr. Azzopardi said there was no record of Mr. Catsimatidis lobbying the governor’s office on the bill.