Rosenstein has been a frequent punching bag for Trump supporters for appointing Special Counsel Robert Mueller in May 2017 to take over a federal probe of suspected Russian meddling in the USA election and potential coordination between Moscow and the Trump campaign. The deputy attorney general has strongly denied those reports.
In a radio interview aired on Monday, Mr Trump said he had not decided whether to fire Mr Rosenstein.
"He's expecting to be fired", a source close to Rosenstein told the publication.
Rosenstein was overseeing Mueller's work after the attorney general, Jeff Sessions, recused himself due to his contacts with Russian officials when he was part of the Trump campaign. He reportedly also whispered to Cabinet officials about invoking the 25th Amendment to remove President Trump from office.
Further, The Washington Post quoted an unnamed individual saying Rosenstein quipped sarcastically after McCabe pushed for an investigation into the president, "What do you want to do Andy?"
This is a breaking news story and will be updated. Trump has repeatedly attacked Sessions for that move. Another official said that Rosenstein made the comments about taping Trump in jest.
Mr. Pompeo, who served as Central Intelligence Agency director before being tapped as the nation's top diplomat, also said he never heard any discussion among Cabinet officials about invoking the 25th Amendment.
But he said when asked about Rosenstein: "I've been pretty clear since the beginning of service here in this administration, if you can't be on the team, if you're not supporting this mission, then maybe you just ought to find something else to do".