He was upset that the reporter, Ryan Lizza, had tweeted about who Mr Scaramucci was dining with.
The former Wall Street financier also attacked Trump aide Steve Bannon.
The conversation with Mr Lizza was littered with profanity-laced insults directed at Mr Priebus and Chief Strategist Steve Bannon.
Mr Scaramucci later tweeted his regret at the "colourful language". He did not deny the nature of the conversation as it was reported but said he "made a mistake in trusting a reporter".
"It won't happen again," he tweeted.
Mr Scaramucci has spent much of his first week in the White House railing against "leakers" and hinted in interviews that he thinks the chief of staff is one of them.
I made a mistake in trusting in a reporter. It won't happen again.
"If Reince wants to explain that he's not a leaker, let him do that," he told CNN in a phone interview.
President Trump has said that leaks from the White House are a national security threat and must stop.
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