When asked if the Trump's tweets should be considered official statements from the White House, Spicer appeared incredulous.
The press secretary's striking refusal to reaffirm the President's confidence in his attorney general came as reports surfaced about the President's lingering frustrations with his attorney general's recusal and a day after Trump took to Twitter to critique his own Justice Department, which is led by Sessions.
Trump has also grown frustrated with the way his travel ban has been stalled in federal courts, the people added.
It is unclear when Sessions offered to resign, and Trump refused the offer.
Despite multiple reports claiming that Sessions had floated the idea of quitting, he will not be stepping down from his post, CNN senior White House correspondent Jim Acosta recently tweeted. Trump made clear to Sessions that while he did not like Sessions' decision to recuse himself, these people said, he still had faith in his attorney general.
Frustration that " runs both ways" may have prompted the suggestion from Sessions that he resign, unnamed sources told ABC.
After the election, Sessions was rewarded with one of the most prominent positions in Trump's new administration, atop the Justice Department. Trump seemed to blame the DOJ for the "politically correct" revised version of the order, which Trump himself signed, posting that the Justice Department "should have stayed with the original Travel Ban, not the watered down, politically correct version they submitted" to the Supreme Court.
Trump's relationship with Sessions, though, is much stronger and longer-standing than his relationship ever was with Comey.
Spicer last month declined to say whether Trump had confidence in his FBI Director James Comey hours before Comey was suddenly fired.
Trump and Sessions share a similar world view on key issues, including crime and immigration.