Sessions bowed out of any involvement in those probes because of his failure during his confirmation hearings to disclose his contacts with Russian Ambassador Sergey Kislyak, saying he "did not have communications with the Russians."
Deputy Attorney General Rod Rosenstein currently oversees the investigation into Russian meddling during the election due to Sessions' recusal. He appointed special counsel Robert Mueller to lead the probe, which Trump has increasingly railed against in public.
The Justice Department declined CNBC's request for comment.
Republicans in both the Senate and the House of Representatives have argued Trump may not be able to find a replacement for Sessions.
Sen. Susan Collins, R-Maine, said last week, "I don't see the president being able to get someone else confirmed" if he fired Sessions.
She made the comments shortly after an excerpt had aired from Trump's Fox News interview, in which the president again griped about Sessions' recusal. He "took the job and then he said, 'I'm going to recuse myself,'" Trump said. "I said, What kind of a man is this?'"
Without explicitly calling out Trump, Sessions fired back against the president in a statement released shortly after the interview aired.
"While I am Attorney General, the actions of the Department of Justice will not be improperly influenced by political considerations," Sessions said.
Senate Majority Leader Mitch McConnell, R-Ky., has also come to Sessions' defense, telling reporters on Tuesday that he has "total confidence" in Sessions.
contributed to this report.