Barack Obama sought to distance the White House from the fierce dispute between top senators and the Central Intelligence Agency on Wednesday, claiming it would be inappropriate for his administration to become involved the clash over an investigation into the use of torture in post-9/11 interrogations.

In the president’s first remarks about the dispute since Dianne Feinstein, the chairwoman of the Senate intelligence accused the CIA of a cover-up and intimidation directed at her staff, Obama said it was not a matter for the White House to “wade into at this point”.

Obama’s remarks are likely to anger Democratic senators on the committee, who have been publicly calling on the president to get involved in the controversy, which has been characterised by bad feeling on both sides.

In a remarkable speech on Tuesday, Feinstein said the CIA had conducted potentially unconstitutional and illegal searches of computers of congressional staff who were conducting an investigation of the agency’s interrogation and detention program.

She called the dispute “a defining moment for the oversight role of our intelligence committee”, while another veteran Democrat, senator Patrick Leahy, later described her speech as the most important any senator had made to the house.

For its part, the CIA has accused Senate staffers of conducting potential criminal activity, and the Department of Justice is conducting an investigation into all the claims.

Amid the lurid allegations, he White House has been seeking to play down the crisis. Earlier on Wednesday, the White House’s press secretary, Jay Carney, characterised the unprecedented clash between the CIA and the Senate as an issue relating to “occasional disputes over protocol”.

Obama cancelled the CIA’s controversial interrogation program – which independent analysts say included techniques that amounted to torture – soon after coming into office, and he has repeatedly promised to declassify parts of the Senate committee’s 6,300-page report.

However the White House finds itself in an awkward position in what is now an open warfare between Obama’s hand-picked CIA director, John Brennan, and Feinstein, one of the administration’s closest allies in the Senate.

In remarks at the White House on Wednesday disseminated via a pool reporter, Obama said he was “absolutely committed to declassifying that report” as soon as it is completed. Feinstein said the committee report was effectively completed in December 2012, but is awaiting amendments, partly based on feedback from the CIA which the committee believes in misleading.

“With respect to the issues that are going back and forth between the Senate committee and the CIA, John Brennan has referred them to the appropriate authorities and they are looking into it and that’s not something that is an appropriate role for me and the White House to wade into at this point,” Obama said.

The president also suggested the CIA’s interrogation program, rather than the dispute with the Senate, remains “the substantive issue”.

However it is not clear how long the White House can avoid becoming embroiled in a bitter, long-simmering disagreement between the CIA and Senate committee, which its staff have in the past had to mediate.

On Wednesday, Carney revealed that the CIA’s top lawyer, who filed an official criminal complaint about Senate aides to the Justice Department, informed White House attorneys in advance. Carney described the notification as “a heads-up” and said the White House did not intervene. “There was no comment, there was no weighing in, there was no judgment,” he said.

However in her speech on Tuesday, Feinstein repeatedly referred to White House involvement in the affair, including an incident in mid-2010, when the CIA claimed – erroneously, the chairwoman said – that it has been ordered by the White House to remove documents from computer drives being used by congressional staff working on the inquiry.

Almost 1,000 classified documents disappeared from the computers used by the Senate aides, who were working out of a facility run by the CIA, and the issue was only resolved after meetings with the then White House counsel in May 2010.

“The matter was resolved with a renewed commitment from the White House Counsel, and the CIA, that there would be no further unauthorized access to the committee’s network or removal of access to CIA documents already provided to the committee,” Feinstein said.

• This article was amended on 14 March 2014. The original referred to an incident in mid-2010 as an incident in mid-2000. This has been corrected.