BEDMINSTER, N.J. — President-elect Donald J. Trump on Saturday moved to mend fences with political rivals after a divisive campaign, meeting with Mitt Romney, who had scathingly criticized him during the race as “a phony” and “a fraud,” to discuss naming him as secretary of state.

The outreach signaled a change in tone one day after Mr. Trump moved to elevate hard-liners to pivotal national security positions. It was not clear whether Mr. Trump had offered the State Department post to Mr. Romney, or whether Mr. Romney, who has broken sharply with him on Russia, free trade and other issues, would accept if he did.

But some strategists argued that merely by reaching out to Mr. Romney, Mr. Trump was demonstrating an openness to new people and ideas, even from the unlikeliest of sources. It may also have been intended to inject the sort of unpredictability and spectacle into the transition process that the president-elect thrives on.

During a weekend of transition talks at Trump National Golf Club here in Bedminster, Mr. Trump was scheduled to hold a series of discussions with what his aides described as a diverse array of potential advisers. The conversations were aimed at showing that the president-elect was willing to look beyond his loyal inner circle to fill his administration.

Among the others who sat down with Mr. Trump were Michelle A. Rhee, a Democrat who served as the chancellor of public schools in the District of Columbia from 2007 to 2010; Robert L. Woodson, an African-American conservative who works on community-based anti-poverty programs; James N. Mattis, a retired Marine Corps general who headed the United States Central Command and is being considered for the post of defense secretary; and Todd Ricketts, an owner of the Chicago Cubs.

Mr. Trump met with Mr. Romney for about an hour and a half. Afterward, both men exited the clubhouse and shook hands for the cameras. “Went great,” Mr. Trump said, cupping his hands at his mouth to project his voice. Mr. Romney then briefly addressed reporters, declining to say whether he was interested in a cabinet position.

“We had a far-reaching conversation with regard to the various theaters of the world with interest to the United States of real significance,” Mr. Romney said. “We discussed those areas and exchanged our views on those topics. A very thorough and in-depth discussion over the time we had. I appreciate the chance to speak with the president-elect and look forward to the coming administration.”

But if he took a cabinet post, Mr. Romney could serve as a moderating influence on the hard-liners Mr. Trump has already selected, including Representative Mike Pompeo of Kansas as C.I.A. director and Stephen K. Bannon as chief strategist. It could also force Mr. Romney to defend administration policies he did not believe in.

John Feehery, a Republican strategist, said Mr. Trump was showing “great magnanimity” by talking to Mr. Romney, the party’s 2012 nominee and a former governor of Massachusetts. “I think it is meant to reassure some of the establishment that he is going to reach out to them, and that’s an important part of healing the party.”

Choosing Ms. Rhee as education secretary “would show real disruption,” Mr. Feehery added. But she is “also someone who has a real track record of delivering the kind of results he’s looking for, and who the establishment hates.”

Ms. Rhee governed with a brash style as chancellor of Washington’s public school system, which was struggling at the time to improve underperforming schools and reverse below-average test scores. She enraged teachers’ unions by firing teachers who had received poor evaluations, renegotiating teacher contracts to weaken seniority protections and tie their pay to student achievement, and endorsing vouchers to allow poor students to attend private schools.

Contrary to Mr. Trump, who has said he will scrap the set of education standards known as Common Core, Ms. Rhee, who founded the education advocacy group StudentsFirst, supports the program.

Mr. Trump traveled on Friday by motorcade to Bedminster. The trip was the second time that Mr. Trump has left Manhattan since Election Day. The other trip was to Washington, where he met with President Obama and congressional Republicans.

The country club is about an hour’s drive from Manhattan and on a rural stretch. Since buying the property in 2002, Mr. Trump has built two 18-hole golf courses on it.

In the evening, Mr. Trump left the clubhouse with Mr. Pence to attend meetings on another part of the property. About an hour later, Mr. Trump returned to the clubhouse to have dinner with Patrick Soon-Shiong, a billionaire pharmaceutical entrepreneur best known for developing the cancer drug Abraxane.