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Two people familiar with the matter say Trump's White House chief of staff, Reince Priebus, and Obama's former White House chief of staff, Denis McDonough, have spoken since Trump claimed, without evidence, that Obama had him wiretapped.

There have also been conversations between other former Obama officials and Trump officials since Saturday.

"There is a dialogue," one person familiar with the conversations said, speaking on condition of anonymity because they are not authorized to speak publicly about the talks.

The question is whether these talks among advisers will ultimately lead to a conversation between the 44th and 45th presidents.

Spokesmen for Trump and Obama declined to comment.

Obama was irked and exasperated in response to his successor's uncorroborated wiretapping accusation, sources close to the former president tell CNN, though these sources say Obama's reaction stopped short of outright fury.

Obama and his aides responded with disbelief when they learned of Trump's Saturday morning tweets laying out the charges. Later in the day, an Obama spokesman said "neither President Obama nor any White House official ever ordered surveillance on any US citizen. Any suggestion otherwise is simply false."

Obama's loyal army of supporters have been far more active in voicing their dissatisfaction with Trump. On social media and television, former aides have been aggressively pushing back on Trump in the first weeks of his presidency.

Presidents Trump and Obama have not spoken since Inauguration Day, when Obama welcomed Trump for coffee in the White House and accompanied him to the US Capitol for the swearing-in ceremony.

The two men had developed what Trump termed a "warm" relationship in the run-up to Trump's inauguration, fostered by an in-person meeting in the Oval Office and several phone conversations.

But people close to both men acknowledge that the bitterness of the presidential campaign, paired with Trump's longstanding antagonism toward Obama regarding his birth certificate, would make a close relationship improbable.

On the weekend that Trump levied his explosive charges, Obama and his wife, Michelle Obama, were spotted at the National Gallery of Art in Washington on a private tour of artist Theaster Gates' new exhibition. The President was all smiles when he departed the museum, dressed casually and carrying a bag from the gallery's gift shop.

Asked Monday whether Trump's claims would damage the relationship between the 44th and 45th presidents, White House press secretary Sean Spicer downplayed any tensions.

"I think that they'll be just fine," Spicer said.

CNN's Kate Bennett contributed to this report.

This story has been updated to reflect CNN's latest reporting.

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Trump, Obama haven't spoken since inauguration, but advisers ...