Correction: An earlier version of this article incorrectly said that the last former U.S. secretary of state to pursue the presidency was James Buchanan. It was Alexander Haig. This version has been corrected.

While former secretary of state Hillary Rodham Clinton sought to ease tensions with President Obama on Tuesday, their disagreements on foreign policy have highlighted a little-noticed aspect of the president’s tenure: He has devoted almost no effort to grooming a successor.

Unlike some other presidents, Obama has not made it a priority to have his ticket mate succeed him, and it’s unclear whether Vice President Biden will seek the Democratic nomination in 2016.

That has left an opening for Clinton, who has stepped firmly into the vacuum this summer, asserting herself as the next Democrat in line to lead the country. But in making the case, Clinton has reignited a debate over what the party stands for at a time when her former boss remains in the White House.

“It’s almost always a complicated process, in the best of circumstances,” said William Galston, a senior fellow in governance studies at the Brookings Institution.

The issue between Clinton and Obama arose over the weekend, when Clinton drew a sharp distinction between her approach to international relations and a slogan the president started using this spring to describe his determination to avoid costly errors: “Don’t do stupid s---.”

Although some Democrats applauded Clinton’s critique — “ ‘Don’t do stupid stuff’ is not an organizing principle” — it prompted a backlash from liberals and Obama loyalists such as former senior White House adviser David Axelrod. On Tuesday, Axelrod took direct aim at Clinton for her vote in 2002 to authorize the U.S. invasion of Iraq.

“Just to clarify: ‘Don’t do stupid stuff’ means stuff like occupying Iraq in the first place, which was a tragically bad decision,” Axelrod tweeted.

Later, Clinton spokesman Nick Merrill issued a statement saying that the former secretary of state had called the president to discuss the interview and “to make sure he knows that nothing she said was an attempt to attack him, his policies or his leadership.”

Referring to the “honest differences” the two had on issues such as Syria during their time serving together, Merrill added: “Some are now choosing to hype those differences but they do not eclipse their broad agreement on most issues. Like any two friends who have to deal with the public eye, she looks forward to hugging it out when they see each other tomorrow night.”

Obama and Clinton are set to attend a party Wednesday on Martha’s Vineyard, Mass., for Ann Jordan, the wife of longtime Democratic adviser Vernon Jordan.

The flare-up is not only a departure from the detente that the Obama and Clinton camps have enjoyed for years; it marks the first time in nearly a decade and a half that an administration has faced the prospect of running the country while one of its own campaigns in part on what it’s doing wrong.

President George W. Bush did not face this predicament because, from the beginning, Vice President Richard B. Cheney had ruled out the idea of seeking higher office. And although Vice President Al Gore distanced himself after President Bill Clinton admitted to having an affair with White House intern Monica S. Lewinsky, he still had the support of a commander in chief who was deeply invested in his succession.

Other presidents — including Dwight D. Eisenhower and Lyndon B. Johnson — had more fraught relationships with their No. 2s as they sought to move up.

In the case of Bill Clinton and Ronald Reagan, Towson University political science professor Martha Joynt Kumar noted, “the strongest candidate” to succeed them was their vice president.

“There’s a different situation here,” she said, adding that although Biden is “a very likeable guy,” Hillary Clinton appears better poised to win the presidency in 2016. “Obama has a stake in the strongest Democratic candidate getting nominated, because a lot of what he has done, for its permanence, is going to depend on a Democrat getting elected.”

Rather than focusing on a direct successor, Obama has concentrated on reshaping the coalition that undergirds the Democratic Party. Although the new demographic alliance he has assembled — which consists of Latinos and young people as well as traditional party stalwarts such as union members and African Americans — will be critical to any Democrat winning the White House in two years, it leaves the broader vision of the party in doubt.

Unlike several other presidents, Obama did not spend an extended period of time building up the state and national infrastructure that typically serves as the launching pad for ambitious politicians.

And the president’s campaign operation, which has evolved into the group Organizing for Action, has operated relatively independently of the Democratic Party. The group works on campaigns to advance the president’s legislative and policy agenda, rather than actual elections.

Faced with the prospect of Republicans winning control of the Senate this fall, Obama has worked to raise money for Democrats through a series of fundraisers. On Monday night, for example, he headlined an event on Martha’s Vineyard for the Democratic Senatorial Campaign Committee at the home of Roger H. Brown and Linda Mason, who founded the child-care provider Bright Horizons. Keenly aware of what GOP gains in the midterm elections could mean for the rest of his presidency, Obama has spoken of 2014 as his “last campaign.”

But raising money for Democratic candidates, some of whom hail from red states and are wary to appear with the president in public, is separate from laying the party’s future direction.

Part of the challenge for Clinton — who would be the first former secretary of state to pursue the presidency since James G. Blaine — is that only twice in the past 66 years has a two-term president been succeeded by a member of his own party.

Aaron David Miller, author of the upcoming book “The End of Greatness: Why America Can’t Have (and Doesn’t Want) Another Great President,” said that if Clinton ends up succeeding Obama, it will be partly “because she’s viewed largely as an extension of some of his policies, but also because the public has tired of Barack Obama and they’re looking for another Democrat who has a different approach.”

He added, “That’s the real question: How different is she really?”