"We're going to give Democratic voters the opportunity to weigh in. But certainly somebody who claims a majority of the pledged and superdelegates, you know, has a strong case to make" in terms of claiming the mantle of the Democratic Party's nomination, he added.

"This is rooted largely in his own judgment," Earnest said, adding that "the Democratic officials and the candidates themselves will have to make their own decision about which metric they're going to follow" in declaring who has secured the Democratic presidential nomination.

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Obama plans to initially focus on groups where he continues to remain popular – young voters and African American voters – and in several key states, according to an administration official who asked not to be identified in order to discuss internal deliberations. In the case of millennial voters, a major trouble spot for Clinton during this year's primary, Obama is more popular now than when he won re-election in 2012.

The president would also likely campaign hard for Clinton in several industrial Midwest states, including Michigan, Wisconsin and Minnesota, where she has underperformed compared to her chief rival, Sen. Bernie Sanders (Vt.).

Earnest noted that Clinton had served in the Cabinet, while Sanders had not. "So the president has gone to great lengths to try to be hands-off to give Democratic voters the opportunity to make a decision about who should represent our party in the general election."

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Sanders and President Obama spoke by phone on Sunday while the senator was campaigning in California, according to two people familiar who requested anonymity to discuss private conversations.

The call came to Sanders while his motorcade was on a highway in the Los Angeles area, hours before a San Diego rally; Obama was in Florida at the time.

Sanders' motorcade pulled over on the side of the highway for him to take the call, and during the conversation the candidate paced outside his SUV with his hand to his ear. One person familiar said he wanted to be outside and alone for the call rather than driving and surrounded by advisers.

Neither the Sanders campaign nor the White House would discuss the conversation.

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Asked about the timing of the endorsement Monday while campaigning in Compton, Calif., Clinton told reporters "that's up to the president," but "we'll be talking about all of that in the next days, and I look forward to that."

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"Obviously I’m excited about having the president’s support, because I have said throughout this campaign, I was honored to serve in the president’s Cabinet as his secretary of state," she said. "I look forward to campaigning with the president and anyone else."

The New York Times reported Monday morning that the president could endorse Clinton in the near future.

Obama is eager to hit the campaign trail, according to several aides, and has already weighed in multiple times on the race to succeed him.

"So the president intends certainly through the fall, if not earlier, to engage in this campaign, and engage in this debate about the future of our country," Earnest said. "And that’s an opportunity the president relishes."

Asked by CBS News' Mark Knoller whether Obama was looking to pick a fight with Republicans, Earnest replied, "Well, he certainly is not going to shrink from it, I’ll tell you that."