Giuliani, 35, told ABC News said last month that he was "strongly considering" a run and that he would be meeting with New York Republican county chairs in the coming weeks.
He also posted a video on Twitter that assailed Cuomo for, among other things, barring gatherings and closing businesses and schools as the coronavirus began killing thousands of New Yorkers last March.
"At the first sign of a problem, they chose to shut us down," he said. "They take away our freedoms. They quarantine healthy people."
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Giuliani served as a White House aide under former President Donald Trump, but he has never run for public office.
The New York governor's race has gotten more convoluted with all the controversy surrounding Cuomo, who has been expected to run for a fourth term but has been accused by multiple women of sexual harassment in the workplace.
Giuliani was hired in March by the conservative network Newsmax as a contributor.
He was a young child when his father became mayor.
At the elder Giuliani's 1994 inauguration, the then-7-year-old joined his father at the podium, mimicking his gestures and repeating some of his words.
The incident was played for laughs in a "Saturday Night Live" skit starring Chris Farley.
Giuliani was recruited for the golf team at Duke University but sued the university in 2008, saying he had been improperly cut from the team.
Duke said the cut was based on bullying behavior, which Giuliani denied. The lawsuit was dismissed in 2010.
Giuliani was present last month when federal agents raided his father's Manhattan home as part of an investigation into the business dealings of the elder Giuliani during his time as Trump's personal lawyer.
The younger Giuliani told reporters the raids on his father's home and office were "disgusting" and "absolutely absurd."
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To win the Republican nomination in 2022, he will have to get past two far more seasoned politicians already in the race.
Zeldin is a well-funded Trump standard-bearer on Long Island, who not coincidentally was endorsed by former Secretary of State Mike Pompeo Tuesday.
"Serving as Secretary of state, I've seen how weak leaders like Andrew Cuomo take advantage of people and of power," Pompeo said. "Cuomo has abused his office to pass punishing high taxes and erode public safety, and now New Yorkers are fleeing the state. That's why I'm supporting my good friend Lee Zeldin for Governor. Lee, like me, is a soldier and I've known Lee a long time. He's a family man who will step up and defend New York against the radical left."
Astorino is the former Westchester County Executive who was the Republican nominee in 2014, losing to Cuomo by about 14 percentage points.
Cuomo, who is facing allegations of sexual harassment and is being investigated over his administration's handling of data on deaths of nursing home residents from COVID-19, has not officially announced whether he is running for a fourth term.
(The Associated Press contributed to this report.)
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