The Sex and the City actor Cynthia Nixon has said “a lot of people” would like her to run for governor of New York, but would not say if she planned to challenge incumbent Andrew Cuomo in next year’s Democratic primary.

Appearing on NBC’s Today show on Tuesday, the actor, who is a vocal critic of Cuomo, would not answer when asked whether she planned to mount a campaign, but said she had heard “rumours” about her potential bid to unseat him.

“I think there are a lot of people who would like me to run, and I think for a variety of reasons,” she said. “But I think the number one is education. We’ve got a real problem on our hands in New York state.”

She went on to criticize Cuomo’s handling of public education, saying the gap between rich and poor schools had never been greater.

Cuomo has been mentioned as a possible White House candidate in 2020, though he says he is planning to seek a third term as governor in 2018.

Rich Azzopardi, the governor’s spokesman, noted that state funding for schools was at an all-time high under Cuomo and 70% of it went to “high-needs” schools. He extended an open invitation to Nixon to meet with Cuomo.

“The more people talking about issues that make New York state stronger, the better,” he said. “We know Ms Nixon is a passionate advocate for education and we would be happy to sit down with her any time to talk about it.”

Nixon has been active in New York City politics, particularly in education advocacy, and has supported its Democratic mayor, Bill de Blasio.

Her name emerged as a potential candidate for governor last week after liberal activists said she would be a good choice.

She noted on Today that she has three children who have attended New York public schools.

Associated Press contributed to this report