The move will spark suspicions of a "whitewash", Neil Kaytal told Katie Tur on MSNBC.
Mueller on Friday released his report to Trump-appointed Attorney General William P. Barr, who on Sunday summarized for the Senate Judiciary Committee that Mueller's investigation did not find that the Trump campaign conspired or coordinated with Russian Federation in an effort to influence the 2016 US presidential election.
But Barr said that the evidence outlined in Mueller's report "is not sufficient to establish that the President committed an obstruction-of-justice offence".
Trump's business, his charity and presidential transition operation remain under investigation, said Sen.
On questions about the potential obstruction of justice by the president, the report "does not conclude that Trump committed a crime".
Democrats are preparing to fight to make sure as much as possible of Mueller's work is released, including Mueller's description of where Trump hindered his investigation.
Listen to the interview in the video above.
US media reports, citing Justice Department sources, have said that no further indictments are expected to come directly from the Mueller probe.
They also raised questions about Barr's conclusion on obstruction of justice, citing a memo he sent to the Justice Department before his confirmation arguing it would be "fatally misconceived" that Trump's firing of FBI Director James Comey could be the basis of an obstruction of justice inquiry.
There will be no additional indictments from Mueller's office, although other prosecutors could use evidence uncovered by the special counsel's team in their own lawsuits.
Key lawmakers, opposition Democrats and some of Trump's Republican allies, have all called for release of the full report, but it is not clear whether Barr will do so. "The findings of the Department of Justice are a total and complete exoneration of the president of the United States".
Mr Trump's former national security adviser Flynn, his former personal lawyer Cohen and his campaign chairman Manafort are among the 34 individuals already indicted by Mr Mueller, but they will be the last, according to Mr Barr.
"Special Counsel Mueller clearly and explicitly is not exonerating the President, and we must hear from AG Barr about his decision making and see all the underlying evidence for the American people to know all the facts".
Trump, however, is celebrating because he has maintained that the charges against him - that he colluded with the Russians to influence voters in the U.S. - were fabricated and part of a larger conspiracy to undermine his win over Democrat candidate Hillary Clinton in 2016. Mueller, a former Federal Bureau of Investigation director, did not interview Trump in person for his probe.
Democrats are reminding that the House voted almost unanimously, 420-0, to release the full Mueller report, which they say is more important not than ever.
Many pointed out that Mueller's report doesn't exonerate Trump - instead, it seems to be leaving that question to Congress.
Democrat Jerry Nadler, chairman of the House Judiciary Committee, announced that he would be calling Barr to testify in the near future.