Then-FBI Director James Comey later disclosed publicly that the FBI was investigating Russian meddling in the 2016 campaign, and whether members of the Trump campaign colluded with Russia on that effort.
Spicer was asked about Trump's tweeting and whether it did more harm than good.
Sessions' decision to recuse himself from his post was based on a Washington Post report that revealed Sessions had met with Russian Ambassador Sergey Kislyak during Trump's presidential campaign and how he neglected to mention that fact during the Senate Judiciary Committee's January confirmation hearing.
Sessions' recusal "has thrown a lot of this into a tail spin", the official said, setting off a chain reaction that eventually led to the appointment of a special counsel, former FBI Director Robert Mueller, to oversee the FBI's Russian Federation investigation.
CNN has previously reported that Trump was frustrated with Sessions' decision to recuse himself.
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When asked if the Trump's tweets should be considered official statements from the White House, Spicer appeared incredulous.
Still, tensions linger. On Monday, Trump derided the revised travel ban and criticized his own Justice Department's handling of the controversial case.
Sessions' offer to resign was first reported by ABC News. Justice Department spokeswoman Sarah Isgur Flores, who was with Sessions in Atlanta Tuesday, declined to comment.
Trump went on to tweet that the people of America need a "tougher" version of the travel ban and not the "politically correct" version drawn up by the DOJ.
Pressed repeatedly on the topic, Spicer said he simply had not talked to Trump about the matter and added: "If I have not had a discussion with him about a subject I tend not to speak about it".