Berman said that until a presidentially appointed nominee was confirmed by the Senate, the office's "investigations will move forward without delay or interruption".
But it was not clear whether Mr Barr can force Mr Berman out.
In the House, Representative Gerry Connolly of Virginia, a member of the Oversight Committee, said the Justice Department owes the Congress an explanation for this "constitutional crisis".
A senior department official said Clayton was planning to leave the administration, wanted to move back to NY and expressed interest in the Southern District position, and Barr thought he would be a good fit.
In the interview, Bolton said of the December 2018 exchange, "it did feel like obstruction of justice to me".
House Judiciary Committee Chairman Jerrold Nadler, also a Democrat, said he intends to invite Mr Berman to testify.
Berman was appointed by then-Attorney General Jeff Sessions in January 2018, months after Bharara was sacked after refusing to resign along with dozens of other federal prosecutors appointed by US President Barack Obama.
Berman's office prosecuted Trump's former attorney and "fixer" Michael Cohen and Rudy Guiliani associates Lev Parnas and Igor Fruman. The company has denied wrongdoing while resisting efforts by prosecutors to testify in a US court.
US ‘To Allow Companies To Work With Huawei On 5G Standards’
The truth is that the Chinese manufacturer has placed the USA at a disadvantage in the standard-setting. Chinese manufacturing giant, Huawei, has been at loggerheads with the USA on multiple issues.
"The President, his associates, there may be anger about the way that some of the prior investigations were conducted, with respect to his former lawyer Michael Cohen and others", said Bharara.
But the Justice Department's Office of Legal Counsel argued in a 1979 opinion that the "power to remove a court-appointed USA attorney rests with the president".
His statement came hours after Attorney General William Barr said Berman was stepping down from his position. Bharara, an outspoken critic of the president, said the timing of the push to replace his successor was odd.
Attorney General William Barr informed the District Attorney for the Southern District of New York (SDNY) Saturday that President Trump has removed him from his post, a day after the federal prosecutor refused to step down. Bharara wrote on Twitter.
While the Senate considers Clayton's nomination, Trump has appointed Craig Carpenito, now the US attorney for the District of New Jersey, to serve as the acting USA attorney for the Southern District, Barr said in his statement. A spokesman for Berman did not immediately comment.
Trump also appointed Craig Carpenito, now the U.S. Attorney for the District of New Jersey, to serve temporarily in the position Berman held while the Senate considers Clayton's nomination. Under federal law, a US attorney who is appointed by district court judges can serve "until the vacancy is filled".
Barr said in his letter that the deputy USA attorney, Audrey Strauss, would take over the top job until a permanent successor is in place.
Before Clayton was nominated to the SEC post by Trump, he was a corporate lawyer at Sullivan & Cromwell.
Barr announced that the president's nominating Jay Clayton, now chair of the SEC, to replace Berman, and that Craig Carpenito, the USA attorney in New Jersey, would serve in the position on an acting basis in the interim.