A fight brewed between Democrats and Republicans over the public release of special counsel Robert Mueller's report on Russian meddling in the 2016 U.S. election, while President Donald Trump kept up attacks on his critics on Monday.

As the Senate Judiciary Committee's Republican chairman called for an investigation into the origins of the probe of any Trump campaign links with Russians, the Senate leader blocked a second attempt by Democrats to pass a measure aimed at pushing the Justice Department into full disclosure of the report.

Six committee chairs in the Democratic-led House of Representatives called on U.S. Attorney General William Barr, in a letter seen by Reuters, to release the full Mueller report to Congress by April 2. Earlier this month, the House voted 420-0 in favor of making the report public, with no Republican opposition.

Barr on Sunday released a four-page summary of conclusions of the investigation that detailed Russian interference but cleared the Republican president's campaign team of conspiring with Moscow.

No one outside the Justice Department has yet seen the report, including the White House. The Justice Department has not said whether it will release Mueller's full report, but Barr has said he will be as transparent as possible.

A person familiar with the matter said there were no plans at this time to show the Mueller report to the White House.

Trump on Monday vented his anger at the inquiry and vowed investigations into unnamed political enemies who did "evil" and "treasonous things." The probe left unresolved the question of whether Trump engaged in obstruction of justice.

Senator Lindsey Graham, the Republican Senate Judiciary Committee chairman and a Trump ally, told reporters he would ask for an investigation. Barr told Graham in a telephone call that he would be willing to testify to the panel about the Mueller probe, according to a spokesman for the senator.

Republican U.S. Senate Majority Leader Mitch McConnell, noting that it took nearly two years for Mueller to conduct his investigation, said after blocking the Democratic measure: "It's not unreasonable to give the special counsel and the Justice Department just a little time to complete their review in a professional and responsible manner."

Trump said on Monday that "it wouldn't bother me at all" if the report were released but that it was up to Barr.

The end of the Mueller inquiry did not spell the end of the investigative pressure on Trump by Democrats, who gave no indication of easing up on their multiple congressional investigations into his business and personal dealings.

Democratic lawmakers were likely to face a protracted legal battle that will turn on Trump's right to keep communications with his advisers private, legal and political experts said.

One of Trump's lawyers, Jay Sekulow, said at least part of the report should be withheld.

Sekulow said it "would be very inappropriate" to release the president's written answers to questions posed by the special counsel, calling the responses provided in November confidential. After lengthy negotiations, Trump reversed his previous stance that he would be willing to submit to an in-person interview with the Mueller team, ultimately agreeing only to provide written answers.