In March Sessions chose to recuse himself from the Trump-Russia probe, which is also looking into allegations that the president's campaign colluded with Moscow, among other things.
Spicer's remarks came after The New York Times reported that Trump had vented intermittently about Sessions since the attorney general recused himself from any Russia-related investigations conducted by the Department of Justice.
White House spokeswoman Sarah Huckabee Sanders did not respond to a request for comment about Sessions' resignation offer.
"I have not had that discussion with him", Spicer told reporters during a White House briefing, adding: "if I haven't had a discussion with him about a subject, I tend not to speak about it".
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It is unclear when Sessions offered to resign, and Trump refused the offer.
Attorney General Jeff Sessions and President Trump attend the 36th annual National Peace Officers Memorial Service on May 15 on Capitol Hill in Washington.
Trump has also attacked London Mayor Sadiq Khan, calling him "pathetic" in the wake of the London terror attack, as well as Qatar, hinting the Muslim country may have sponsored radical ideology. Trump apparently believes the ultimate responsibility for Robert Mueller's selection as special counsel belongs to Sessions and his decision to step aside.
"Well, the president is the President of the United States so they are considered official statements by the President of the United States", Spicer said.
CORRECTION: This story has been updated to properly reflect the date Sessions recused himself from the Russian Federation investigation - it was March 2. But the exchange comes amid increasingly outward signs of displeasure from Mr. Trump and the White House regarding Mr. Sessions and the Justice Department more broadly. And Sessions' own team has become a part of Trump's inner circle: former Sessions chief of staff Rick Dearborn is now Trump's deputy chief of staff, and former Sessions spokesman Stephen Miller has evolved into a highly influential figure as Trump's policy director and speechwriter. The president traces a direct line between Sessions' decision to recuse himself from any Russian Federation investigation to where he stands now: With former FBI Director Robert Mueller appointed as a special counsel and with intense public focus on the Russian Federation investigation.