President Trump has no plans to oust special counsel Robert Mueller, the White House said Monday, but Democrats on Capitol Hill said they wanted more assurances now that Mr. Mueller has unveiled charges against three former Trump campaign figures.

Democrats said they’ll push for Congress to pass a law that would prevent the president from acting unilaterally to fire a special counsel. They said given Mr. Trump’s history, they want legal guarantees he can’t mess with Mr. Mueller’s probe.

“The president must not, under any circumstances, interfere with the special counsel’s work in any way. If he does so, Congress must respond swiftly, unequivocally and in a bipartisan way to ensure that the investigation continues,” said Senate Minority Leader Charles E. Schumer, New York Democrat.

Republicans said they couldn’t imagine Mr. Trump would interfere with Mr. Mueller now after he unsealed charges Monday against former Trump campaign Chairman Paul Manafort and his close associate Rick Gates, and announced a guilty plea from former campaign foreign policy adviser George Papadopoulos.

“I don’t think anybody in their right mind at the White House would think about replacing Mr. Mueller,” said Sen. Lindsey Graham. The South Carolina Republican said that included Mr. Trump.

Sarah Huckabee Sanders, the White House press secretary, said there’s no need for concern.

“I’ve said several times before, there’s no intention or plan to make any changes in regard to the special counsel,” she said at the daily press briefing, adding Mr. Trump himself made a similar commitment earlier this month.

The White House distanced itself from the developments, saying the charges against Mr. Manafort and Mr. Gates were for activity that predated the campaign, while Mr. Papadopoulos’ guilty plea was for lying to the FBI rather than campaign behavior.

But Democrats said the details of the charges — which included Mr. Papadopoulos meeting with figures offering “dirt” on Hillary Clinton and learning of thousands of emails Russian-linked operatives had from her — show Mr. Mueller’s probe is zeroing in on the campaign.

“Today’s announcement is not ‘fake news,’ these are real indictments and a significant guilty plea,” said Sen. Jack Reed, Rhode Island Democrat.

The developments dominated news on Capitol Hill, where Republicans holding a press conference on judicial nominations were bombarded instead with questions about the charges.

“I don’t see how the indictment changes the president’s ability to do his job,” said Sen. John Cornyn, Texas Republican.

But he ducked a question about whether the Senate would attempt to pass bills protecting Mr. Mueller or other special counsels from presidential meddling.

Senate Majority Leader Mitch McConnell’s office also didn’t respond to a request for comment on whether those bills would get floor time.

The bipartisan bills would require a panel of judges to approve any attempt by the executive branch to fire a special counsel. Mr. Graham is a co-sponsor of one version, while Sen. Thom Tillis of North Carolina is backing another version.

Mr. Tillis said he has wanted the legislation even before Monday’s indictments, and said he wants it to cover future special counsels, not just Mr. Mueller.

Democrats and liberal activist groups are rallying around the legislation, saying it’s a concrete step Congress can take to protect the integrity of the ongoing investigation.

Multiple congressional committees continue their own investigations into Russia’s efforts to meddle in the 2016 election.

House Minority Leader Nancy Pelosi said she still wants to see yet another investigation launched.

“Even with an accelerating Special Counsel investigation inside the Justice Department, and investigations inside the Republican Congress, we still need an outside, fully independent investigation to expose Russia’s meddling in our election and the involvement of Trump officials,” Mrs. Pelosi said in a statement.

Democrats also said they want to see Congress stiffen disclosure rules for those working on behalf of foreign governments. The indictment against Mr. Manafort includes multiple charges that he worked for pro-Russian interests in Ukraine without properly registering.

⦁ David Sherfinski contributed to this report.

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