The release of the Special Counsel Robert Mueller’s report’s key findings on Sunday could mark the start of a lengthy battle to see the entire Mueller report made public.
A number of senior Democrats have called for the full report to be released along with all of the special counsel’s investigative files.
Mr Barr has said he will release more, but indicated it would take some time to determine what materials could be shared.
He did not give a specific time frame, but whenever further details are handed to Congress Democrats may mount legal challenges if it is anything less than the entire report.
As Congress awaits further details, Mr Barr may be called to testify in front of the House Judiciary Committee.
The committee’s Democratic chairman, Jerry Nadler, said on Twitter that he would ask Mr Barr to testify “in the near future” over “very concerning discrepancies and final decision making at the Justice Department”.
Meanwhile, as Mr Trump claimed “total exoneration” on Sunday, about a dozen other investigations were continuing to examine his activities.
They include a federal investigation in New York that is looking into possible election-law violations by the Trump campaign and his businesses and possible misconduct by the Trump inaugural committee.
Congress is also continuing its own inquiries, mostly in the Democratic-controlled House of Representatives.
Congressman Jerry Nadler, the Democratic Chair of the House of Representatives Judiciary Committee, emphasised the attorney general had not ruled out that Mr Trump might have obstructed justice.
“Barr says that the president may have acted to obstruct justice, but that for an obstruction conviction, ‘the government would need to prove beyond a reasonable doubt that a person, acting with corrupt intent, engaged in obstructive conduct’.”
Democratic Senator Richard Blumenthal, a member of the Senate’s Judiciary Committee, said that while there was a lack of evidence to support “a prosecutable criminal conspiracy”, questions remained over whether Mr Trump had been compromised.
Democratic House Speaker Nancy Pelosi and Senate Democratic Leader Chuck Schumer said in a joint statement that Mr Barr’s letter “raises as many questions as it answers” and called for access to the full report.
“For the president to say he is completely exonerated directly contradicts the words of Mr Mueller and is not to be taken with any degree of credibility,” the statement said.
White House press secretary Sarah Sanders described the findings of the report as “a total and complete exoneration of the president”.
Mr Trump’s lawyer, Rudy Giuliani, said the report was “better than I expected”. Republican Senator Mitt Romney welcomed the “good news”, tweeting that it was now “time for the country to move forward”.