In phone calls last week, New York Lt. Gov. Kathy Hochul sought advice on potential first steps in office, as well as whom to hire and which members of Gov. Andrew Cuomo’s administration might stay on if he resigned or was removed from office, people familiar with the matter said.

Ms. Hochul told people that she believed that as first in the line of succession, she could be called on to take over in a matter of weeks, the people said. If so, she would be the first woman to serve as governor of New York.

Mr. Cuomo is grappling with an impeachment inquiry and a report that concluded he broke state and federal sexual harassment laws.

While she is well-known in western New York, state officials said Ms. Hochul hasn’t been a part of the Democratic governor’s inner circle and didn’t benefit from the national attention he gained during the coronavirus pandemic—nor the criticism he endured as more about his actions surfaced.

Since taking office in 2015, the Democratic lieutenant governor has regularly kept up contact with political and governmental leaders around New York and has traveled to all 62 counties in the state. She previously had served as clerk of Erie County, which includes her home city of Buffalo, and was elected to Congress in 2011 for a partial term. As lieutenant governor, she chairs a group of regional economic-development councils and often presides over the state Senate.

An official in the lieutenant governor’s office said Ms. Hochul was receiving advice from a variety of sources but had no inside information on Mr. Cuomo’s plans. The official said it was the job of the lieutenant governor to be prepared to take over as governor.

Ms. Hochul had been treading lightly since Mr. Cuomo requested in March that Attorney General Letitia James investigate accusations that he sexually harassed multiple current and former state employees. Ms. James said in a report released Tuesday that Mr. Cuomo inappropriately touched, hugged, kissed or commented on 11 women.

The Democratic governor said Ms. James’s review was biased and has denied inappropriately touching anybody. He has rebuffed calls for his resignation from Republicans and prominent Democrats, including President Biden.

Ms. Hochul last week said the report “documented repulsive and unlawful behavior by the governor” and that “no one is above the law.” She didn’t call for Mr. Cuomo’s resignation, but said the state Assembly—which is preparing articles of impeachment—would determine the next steps.

The Assembly judiciary committee is scheduled to meet Monday and offer an update on the impeachment investigation. Lawyers working for the committee gave Mr. Cuomo until Friday, Aug. 13, to submit evidence in his defense. If the Democrat-dominated chamber approves an impeachment charge, Ms. Hochul would become acting governor while Mr. Cuomo is tried before the state Senate and judges of the state’s highest court.

Ms. Hochul didn’t communicate directly with the governor last week, according to people who are close to both of them. The governor’s staff didn’t see or approve her statement before it was issued, the official said.

Rich Azzopardi, a senior adviser to Mr. Cuomo, declined to comment on whether the governor has spoken with Ms. Hochul or about her recent activities.

“ “Not only does she know all the players in the Democratic Party and the government, she’s respected by them.” ”

— New York State Democratic Committee Chairman Jay Jacob

Mr. Cuomo tapped Ms. Hochul in 2014 after his first running-mate, Bob Duffy, asked to leave the ticket so he could return to his home in Rochester. Ms. Hochul was working as a vice president of government relations of M&T Bank after losing her re-election bid to former U.S. Rep. Chris Collins, a Republican who later resigned just before pleading guilty to insider trading charges.

In 2018, Mr. Cuomo publicly suggested that Ms. Hochul leave the ticket and campaign against Mr. Collins after he was charged in the insider-trading scheme. Ms. Hochul declined. She appears infrequently with the governor and remained in Buffalo while Mr. Cuomo managed the state response to the coronavirus last year. In his memoir of the pandemic, Mr. Cuomo didn’t mention Ms. Hochul in the 279-page book.

New York State Democratic Committee Chairman Jay Jacobs said he talked with Ms. Hochul on Wednesday after he called on Mr. Cuomo to resign.

“I don’t think it’s going to take much for her to step up,” Mr. Jacobs said. “Not only does she know all the players in the Democratic Party and the government, she’s respected by them.”

Ms. Hochul canceled a planned press conference in Brooklyn on Wednesday but gave a speech that evening in Harlem to the National Organization of Black Elected Legislative Women, according to state Assemblywoman Alicia Hyndman, who was present. Ms. Hochul spoke about her path to elected office and how she was inspired by Shirley Chisholm, a Black congresswoman from Brooklyn who ran for president in 1972.

Ms. Hochul didn’t mention Mr. Cuomo’s situation, but one attendee from Alabama thanked her and joked that “we could be addressing the next governor,” Ms. Hyndman said.

—Deanna Paul contributed to this article.

Write to Jimmy Vielkind at Jimmy.Vielkind@wsj.com