Adams, a Democrat, has endured months of mounting calls for his resignation or removal and Hochul said she had heard from many New Yorkers who feel "outraged, hurt, betrayed" by the mayor.
The calls have grown louder in recent days as the mayor pledged to help US President Donald Trump, a Republican, with deporting undocumented immigrants while Trump's Justice Department sought to dismiss the federal indictment against Adams.
A US district judge in Manhattan is considering whether to drop the charges.
The state's constitution gives the governor the power to remove a mayor, and to suspend him or her for 30 days once charges have been brought.
The power has not been invoked in nearly a century, and the process has never been carried through to completion.
Hochul said she was "deeply troubled" by the accusations against Adams and said New York City was in crisis, but said on Thursday that she would not begin removal proceedings for now, though she would retain the option to do so at a later date.
"My strong belief is that the will of the voters and the supremacy and sanctity of democratic elections preclude me from any action," she told reporters.
"I cannot deny the people of this great city the power to make this decision for themselves."
Instead, she proposed New York create a new state inspector-general overseeing New York City.
Adams, who is seeking a second term in elections this November, has repeatedly said he has done nothing wrong and will not resign.
He pledged to co-operate with Hochul, saying "I am willing to work with her to ensure faith in our government is strong".
Hochul's plans would need votes of approval the state legislature and the city council, whose members might oppose giving the state greater control over the city.
Adams was indicted in September on five corruption charges, accusing him of arranging illegal foreign donations to his election campaigns and luxury travel from Turkish nationals seeking to influence him.
He has pleaded not guilty.
He has publicly courted Trump, who returned to the White House in January and has the power to pardon Adams.
More than half a dozen senior prosecutors at the Department of Justice have resigned in protest over the effort to have the indictment against Adams dismissed.
