The statement comes as three of Trump's close associates, including his son, have been called to Senate committees to answer questions about alleged Russian meddling in the presidential campaign last year.
Donald Trump said he regretted his choice of attorney general on Wednesday as he sought to protect himself from the growing firestorm over Russia's alleged election meddling which is engulfing his presidency.
On the eve of the six-month mark of Trump's inauguration, it also emerged that senators next week will grill three of the pivotal players in the Trump campaign, including his eldest son, over swirling allegations of the presidential campaign's collusion with Russia.
The announcements came as the Trump administration and the Kremlin tried to quell an uproar over a previously undisclosed meeting between Trump and his Russian counterpart Vladimir Putin during this month's G20 summit.
TRT World's Jon Brain reports.
Attorney General Jeff Sessions was one of the first senior Republican politicians to endorse Trump before last November's election and was rewarded by being appointed America's top law enforcement officer.
But he stood aside in March from overseeing an FBI-led probe into whether members of the Trump team colluded with Moscow during the election campaign after it emerged that Sessions had not disclosed during his Senate confirmation hearing that he met twice with the Russian ambassador to Washington.
In an interview with the New York Times, Trump said Sessions had acted unfairly in taking the job in the first place if he had felt in any way compromised.
"How do you take a job and then recuse yourself? If he would have recused himself before the job, I would have said, 'Thanks, Jeff, but I'm not going to take you,'" Trump said.
"It's extremely unfair – and that's a mild word – to the president."
Trump also criticised Sessions' performance at the Senate confirmation hearing in January, in which he denied meeting any Russians when he had in fact met the Russian ambassador.
"Jeff Sessions gave some bad answers," the president said in his interview with the Times.
"He gave some answers that were simple questions and should have been simple answers, but they weren't."
The Senate Intelligence Committee is now due to grill the president's son-in-law Jared Kushner, who is one of the most powerful figures in the Trump White House, in a closed-door session next Monday, his lawyer told CNN.
And two days later, the Senate Judiciary Committee will question Trump's eldest son, Donald Trump Jr, and former campaign manager Paul Manafort about Russia's role, congressional officials announced on Wednesday.
All three men attended a controversial meeting with a Russian lawyer last year, in which they were expecting to receive dirt from Moscow on Trump's election rival, Hillary Clinton.
Trump Jr sent shock waves through Washington last week by releasing a series of emails that detailed how he had attended the meeting after being promised "very high level and sensitive information" that was "part of Russia and its government's support for Mr. Trump."
Both Trump Jr and Manafort were told in letters from the panel's senior members to "preserve all relevant documents in your possession ... related to Russian interference in the 2016 election, including documents related to your or the Trump campaign's contacts with Russian government officials, associates or representatives."
The senators said they would issue subpoenas if the witnesses did not produce the required documents.
With Washington reeling from the stream of revelations, Trump has rushed to his son's defence, and lambasted what he calls a political "witch hunt."
My son Donald did a good job last night. He was open, transparent and innocent. This is the greatest Witch Hunt in political history. Sad!— Donald J. Trump (@realDonaldTrump) July 12, 2017
But on Wednesday night a senior official made clear that the administration was cooperating with the investigation being conducted by a special counsel, former FBI director Robert Mueller, who is probing the Russia connection independently of the Senate panels.
"I'm part of the Trump administration, and the special counsel is reporting to me, and the answer to that is certainly yes," deputy attorney general Rod Rosenstein told Fox News.
"We are providing appropriate cooperation to make sure the investigation is done properly."
Some critics have seized on the additional, previously undisclosed meeting between the US president and his Russian counterpart as more evidence that the Trump team is reluctant to come clean on anything involving Russia.
The White House confirmed the politically sensitive sit-down over dinner on the final night of the G20 summit in Germany only after it was leaked, but has denied that it amounted to anything untoward.
"Once again, the Russia fever has caught up with the media and everybody ran out and tried to create a story that simply didn't exist," White House spokeswoman Sarah Huckabee Sanders said at a briefing on Wednesday.
That message was echoed by her Kremlin counterpart Dmitry Peskov.
"The use of the terms 'secret' or 'confidential' for this meeting provokes absolute surprise and incomprehension."
The meeting had been "officially accepted" by diplomatic channels, he added in comments reported by the Russia's TASS news agency.