The roster of possible opponents is mostly familiar to the governor, including the man he defeated in 2014, Rob Astorino, the Westchester County executive. But Mr. Cuomo may also be facing fire from the left, and not just from Ms. Miner, who has garnered attention for her willingness to criticize the governor. The Wall Street Journal broke the news of a possible run by the actress and activist Cynthia Nixon last week, as well, though her representatives declined to comment on such speculation.

Terry Gipson, a former state senator and another possible Democratic challenger, said he had been particularly troubled by the governor’s willingness to deal with the Independent Democratic Conference in the Senate. The conference is an eight-member coalition that collaborates with Republicans, allowing that party to control the chamber despite being outnumbered by Democrats.

“There’s been a wide variety of progressive legislation that was blocked by the I.D.C. and that seemed to have no real support from the governor,” said Mr. Gipson, who represented a portion of the Hudson Valley before losing his seat in 2014. “Democrats feel that the state needs a real Democrat to move it forward.”

In 2014, the governor’s progressive bona fides were questioned by activists who felt that his often cozy relationship with Republicans in Albany had come at a cost for liberal policies. Members of the Working Families Party, which sits to the left of most mainstream Democrats and has a tense relationship with Mr. Cuomo, have entertained the idea of a third-party run against the governor next year, either in a primary or general election.

Zephyr Teachout, who used a similar sentiment to mount a surprisingly strong challenge to Mr. Cuomo in 2014, said she believed that a primary was the biggest threat to the governor, as the Democratic base has been fired up “in Trump times.”