Mr Kennedy, who owns three water dogs himself, said in January that he was "proud to endorse the Portuguese water dog as the next First Dog of the United States".
He said the breed would be a "perfect fit for the Obama Family" as the dogs have a "can-do and hopeful spirit" and were smart, resilient, determined, optimistic and tireless.
Bo was reported to have charmed the first family during a secret meeting at the White House a few weeks ago, the Post said, citing an unnamed source.
It said the top dog made no toileting errors and did not gnaw on the furniture because he has been receiving lessons in good behaviour from the Kennedys' dog trainers at a secret location outside Washington.
The choice of the US presidential pooch has been a pet topic for the nation's press ever since Mr Obama promised his daughters a puppy during the campaign.
The president, for his part, has embraced the media buzz.
Obama puppy keenly anticipated
"Oh, man, now, that's top secret," he joked on Friday as reporters pressed him again on the issue.
He also made light about his promise to his girls during last month's appearance on Jay Leno's talk show.
"This is Washington. That was a campaign promise," Mr Obama said, as the audience roared with laughter. "No, I'm teasing. The dog will be there shortly."
But sources told the Post that the dog was not in the White House as of Saturday evening.
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