IIRC Sessions never offered to resign. He said POTUS must fire him to get rid of him. In this climate, firing Sessions would only increase the left’s insanity. Trump must wait at least until after mid-terms, in hopes he picks up larger majorities in both houses.
According to the New York Times, the president was so upset with Robert Mueller being appointed special counsel that he berated Sessions in the Oval Office during a May meeting.White House Counsel Donald McGahn received a call from Deputy Attorney Rod Rosenstein saying he had appointed Mueller to be a special counsel for the investigation. McGahn then told the Trump, Sessions and administration advisers, who were in the process of discussing a replacement for fired FBI chief James Comey.
Trump went on to berate Sessions — who had recused himself from the investigation and thus paved the way for a special counsel — by calling him an "idiot" and saying choosing him to be attorney general was one of the "worst" decisions he had made. He then demanded Sessions resign.
According to the report, Sessions was "emotional" when he left that Oval Office meeting, agreeing to resign. Later that night as the Justice Department publicly announced the appointment of Mueller, Sessions wrote his short letter of resignation.
Meanwhile, Vice President Mike Pence, then-White House chief strategist Steve Bannon and then-Chief of Staff Reince Preibus intervened and convinced Trump to allow Sessions to stay.
Sessions allegedly told his associates following the Oval Office incident "that the demeaning way the president addressed him was the most humiliating experience in decades of public life."
According to the Times, Trump returned Sessions' resignation letter to him — which included a handwritten response.