Scaramucci's initial message appeared to reference a Politico story published Wednesday that revealed details about his financial disclosure form.
Scaramucci's social-media postings come amid long-playing rumors of turmoil with Priebus - and speculation that Priebus' job may be on the line now that Scaramucci is firmly in President Donald Trump's inner circle as the new White House communications director.
Scaramucci has all but officially accused Priebus of being behind the leak of his Office of Government Ethics forms, which showed Scaramucci made at least $10 million past year from his investment firm SkyBridge Capital and still stands to profit from it while working in the White House.
Anthony Scaramucci condemned the financial "leak" after a report by Politico said he stands to make millions from his ownership stake in an investment firm. He later deleted the tweet, apparently after being informed that the form was not leaked.
Trump's first White House press secretary, Sean Spicer, resigned last week with the news that Scaramucci was being brought on as communications director.
Mr Scaramucci also tweeted that it was a "public notice to leakers" that all senior administration officials "are helping to end illegal leaks". The documents are publicly available on request.
CNN contributor Ryan Lizza said Wednesday night that Scaramucci blamed Priebus "for blocking him at every turn" when he tried to get a job in the White House soon after Trump was elected. "We are a little bit like brothers where we rough each other up once in a while, which is totally normal for brothers. When I said we were brothers from the podium, that's because we're - some brothers are like Cain and Abel". But, as NBC News's Peter Alexander said on Thursday, lately has more the feel of "Game of Thrones".
Scaramucci, however, did not go so far as to directly accuse Priebus on CNN, arguing that he meant he is working with senior officials on eradicating the leaking problem.
Priebus has not made any public statements on the matter.
"If you want to talk about the chief of staff, we have had odds, we have had differences", Scaramucci said.
"He's looking at these sanctions right now, he may decide to veto the sanctions and be tougher than the Russians than the Congress. [Crosstalk] He may sign the sanctions, exactly the way they are, or he may veto the sanctions and negotiate an even tougher deal against the Russians". In 2012, he tweeted about meeting Paterno and called him an "honorable man", saying, "I met Joe Paterno twice ('92 & '96) he was an honorable man. I don't know if this is reparable or not - that will be up to the president", he said. "What I'm upset about is the process and the junk pool, the dirty pool ... in terms of the way this stuff is being done, and the leaking won't stop".
Speaking on CNN Thursday morning, Scaramucci was more explicit about his tweet. "I will be contact @FBI and the @TheJusticeDept #Swamp @Reince45".