Harvard’s next president likely won’t be named for months. But speculation is underway.

Almost as soon as Claudine Gay resigned as Harvard University’s 30th president last week, the conjecture began about who might be the 31st.

Harvard has not identified any candidates. In a statement, a university spokesperson said the search for a new leader would begin in due course and “will include broad engagement and consultation with the Harvard community in the time ahead.”

But school officials most likely have ideas already about potential candidates, said Ted Mitchell, president of the American Council on Education, a higher education policy and professional development group.

“Universities always keep track of their faculty members who have gone on to be leaders in other institutions,” he said. “That’s not a formal list in anyone’s desk drawer; it’s kind of just being part of the business.”

On social media, a few suggestions have begun circulating, including Deval Patrick, a former two-term Massachusetts governor, and Danielle Allen, a former Massachusetts gubernatorial candidate and a professor of political philosophy, ethics, and public policy at Harvard. Barack Obama’s name has come up, too.

Gay’s interim replacement, Alan Garber, a Harvard physician and economist, could be considered as well, due to his experience as the college’s provost for the past 12 years.

It’s not an ideal time for a university to search for a new leader, Mitchell said. Harvard joins other prestigious universities, including University of Pennsylvania and Stanford University, looking for presidents in 2024, while the University of California, Los Angeles, is seeking a new chancellor. The number of people qualified and willing is small, Mitchell said, in part because campuses have become a focal point for America’s culture war between conservative and progressive values.

“These are really tough jobs,” he said. “They are only becoming more difficult and more visibly difficult because of the spotlight that’s been put on campuses in the last two to three years.”

Gay ultimately faced scrutiny over accusations of plagiarism, but she first became a lightning rod in the wake of Hamas’s Oct. 7 attack on Israel. Her initial statement, and comments later made to a congressional committee, were perceived as inadequate after the horror of the attacks and amid concerns about antisemitism on campus. Now, students on both sides of the fraught debate over the Mideast conflict say they want a president who will foster an environment that feels safer.

In interviews Friday, faculty, students, and outside observers said they wanted a president with a record of serious scholarship, a knack for communication, and a commitment to academic freedom.

Robert Putnam, professor emeritus of public policy at Harvard, emphasized that he had no insight into the university’s hiring process, but said experience and demographics should be considered.

“There ought to be somebody who has a track record. We’re not going to be guessing whether they’re going to be good or not; we know they’re good,” he said. It would also be “highly desirable that the person be either a person of color, or a woman, or even better, both,” he said.

After the vitriol Gay drew, which some interpreted as particularly vicious because she is a Black woman, that might be difficult.

“It will be harder to build a pool of women and people of color for that role given what happened to Claudine Gay,” said Larry Ladd, senior consultant with the Association of Governing Boards of Universities and Colleges. “The president of Harvard is a symbol as much as anything else. It’s a very symbolic role.”

It will likely take months before the next president is named. The search that ended with Gay’s selectiontook five months, and that was fast by Harvard’s standards.

Obama and Garber did not respond to requests for comment. Putnam, who knows Obama, said he doubted the former president would take the job.

Patrick, now co-director of the Center for Public Leadership at the Harvard Kennedy School, earned undergraduate and law degrees from Harvard and teaches there. An able fund-raiser and gifted orator, Patrick also served in the Justice Department under the Clinton administration and has worked in corporate America, for Texaco and Coca-Cola and for more than four years at Bain Capital.

He declined to comment, and it’s not clear if the Democrat — who four years ago mounted a White House bid — would be interested.

He has told others that he’s grateful to Harvard for his experience there, but doubts that he’s right for the job, according to a person close to Patrick who requested anonymity to discuss private conversations.

Allen downplayed suggestions on social media that she should be considered, writing Wednesday on X, the site formerly known as Twitter, “let’s all just take a breath.”

“I appreciate the kind words out there,” she wrote. “My heart just hurts for Claudine right now. . . . We faculty members have lots of work to do. That’s my only focus.”

Allen did not respond to an interview request Friday, but she wrote about the need for a “renovation” of colleges and universities’ approach to diversity, equity, and inclusion efforts in The Washington Post on Dec. 10, just days after Gay’s congressional testimony.

In the opinion piece, Allen noted she helped lead a task force that in 2018 delivered a report on addressing inclusion on campus. But administrators at Harvard “largely overlooked” several major themes of the report, she said, including the task force’s “focus on academic freedom, on the need to make space for religious identity and on the need for greater political diversity on our campus.”

“We should not just protect students’ speech rights but also insist that they exercise those rights in accordance with campus norms for a culture of mutual respect,” Allen wrote.

Both students who support Israel and those who support Palestinian causes feel that mutual respect has been lacking since the Oct. 7 Hamas attack on Israel.

“When students are calling for violence against Jews, calling for an intifada, attacking Zionism for all the evils in the world, that’s not an international politics issue, that’s a campus issue,” said Charlie Covit, a Harvard freshman involved in Jewish student organizations.

Jana Amin, president of the Society of Arab Students and a junior at Harvard, also felt the university had failed to provide a safe space for Middle Eastern students and students opposed to Israel’s violent response in Gaza to the Hamas attack.

“There have been Arab students wearing Palestinian symbols over the past couple months who faced harassment and intimidation,” she said. “Those have not been taken seriously.”

Mitchell, of the national higher education organization, agreed that Harvard needed a peacemaker.

“Harvard’s challenge has a lot to do with calming the waters, with creating at least a common dialogue between a lot of factions that now appear to be at odds with each other,” he said.

Putnam, the public policy professor, said finding someone brilliant with public relations acumen and experience may seem like a high bar to meet, but he named several people, including Allen, he felt could meet the challenge.

“What I’m really trying to emphasize to you is we can have it all,” he said.

Hilary Burns of the Globe staff contributed to this report.