According to the Politico media outlet, citing sources close to the attorney general, Sessions' offer came just before the first foreign trip of Trump's and came in response to the president's negative reaction to the fact that Sessions made a decision to recuse himself from the investigation into Russian interference.
With the president's frustrations with his Justice Department spilling into public view, top White House spokesman Sean Spicer declined to say Tuesday whether President Donald Trump has confidence in Attorney General Jeff Sessions. "I have not had a discussion with him (Trump) about that", Spicer said.
Sessions recused himself from investigations into Russian Federation meddling in the 2016 presidential election and possible ties between Moscow and Trump's campaign.
Tensions between the two led Sessions to offer his resignation.
Sessions was the first sitting United States senator to endorse Trump and became one of his top advisers on the campaign trail.
Police name two London attackers as Khuram Butt and Rachid Redouane
The attackers, wearing fake suicide vests, were shot dead by eight officers after police opened fire with a hail of 50 bullets. But "there was no intelligence to suggest that this attack was being planned", police said in a statement .
When pressed, Spicer responded: "I don't have a comment about that".
After the election, Sessions was rewarded with one of the most prominent positions in Trump's new administration, atop the Justice Department.
And on Monday, Trump unleashed a series of tweets that accused Sessions' department of devising a "watered down, politically correct" version of the travel ban. "If I haven't had a discussion with him about a subject, I tend not to speak of it".
But the cozy, warm relationship between Trump and Sessions has appeared to turn ice cold, at least for the president, the New York Times reports. Sessions announced his recusal shortly after he became attorney general and a day after The Washington Post revealed that he had twice met with Russian Ambassador Sergey Kislyak during the campaign and did not disclose that fact to the Senate Judiciary Committee during his January confirmation hearing.