President Trump lashed out at special counsel Robert Mueller’s investigation into Russian meddling Sunday, saying it is tainted by political bias and won’t find evidence of collusion with his 2016 campaign — sparking bipartisan fears that he is stress-testing a plan to have the high-profile investigator fired.

Mr. Trump, in a series of online posts, celebrated the firing of FBI Deputy Director Andrew McCabe over actions related to an investigation of the Clinton Foundation. He said Mr. McCabe’s termination supports his belief that federal investigators cannot be trusted and will cook up ways to fault him.

“And yet, there is NO COLLUSION!” Mr. Trump said.

The president also cast doubt on Mr. McCabe’s claims that he kept detailed memos of his interactions with Mr. Trump. He said Mr. McCabe never took notes in his presence and likely crafted documents to fit his own agenda.

His Twitter rant followed his attorney’s call on the Justice Department to shut down Mr. Mueller’s investigation immediately on the basis that it has been tainted by political bias from the beginning.

The rapid escalation of acrimony between the White House and federal law enforcement alarmed lawmakers in both parties, who said firing Mr. Mueller would be a bridge too far and spark a constitutional crisis.

Mr. Mueller has secured guilty pleas from Trump campaign associates and indicted Russians accused of online trolling. Yet the investigation appears to be inching closer to the president, including a subpoena for records from the Trump Organization, which oversees the first family’s business interests.

“The president is floating trial balloons about derailing the Mueller investigation,” said Senate Minority Leader Charles E. Schumer, New York Democrat. “Our Republican colleagues, particularly the leadership, have an obligation to our country to stand up now and make it clear that firing Mueller is a red line for our democracy that cannot be crossed.”

Sen. Jeff Flake of Arizona said he would expect his fellow Republicans to reject attempts to undermine Mr. Mueller.

“It seems to be building towards that. I just hope it doesn’t go there, because it can’t,” Mr. Flake told CNN’s “State of the Union.”

The White House released a statement Sunday night by one of the president’s lawyers, Ty Cobb, refuting reports that Mr. Trump is considering firing Mr. Mueller.

“In response to media speculation and related questions being posed to the administration, the White House yet again confirms that the president is not considering or discussing the firing of the Special Counsel, Robert Mueller,” Mr. Cobb said.

The dispute was ignited by Attorney General Jeff Sessions’ firing of Mr. McCabe late Friday, just two days before the deputy director’s retirement, likely putting at risk the pension he accumulated as a 21-year FBI veteran.

The Justice Department inspector general said Mr. McCabe misled investigators about his role in allowing FBI officials to speak to reporters at The Wall Street Journal about a corruption investigation into the Clinton Foundation.

Mr. Trump applauded the firing as a “great day for democracy.”

Sen. Marco Rubio, Florida Republican, said Mr. McCabe should have been able to secure his full retirement because an inspector general’s report on the episode hasn’t been released and the fired deputy said Mr. Trump scapegoated him as part of an effort to discredit Mr. Mueller’s investigation.

“This attack on my credibility is one part of a larger effort not just to slander me personally, but to taint the FBI, law enforcement and intelligence professionals more generally,” Mr. McCabe said in a statement. “It is part of this administration’s ongoing war on the FBI and the efforts of the Special Counsel investigation, which continue to this day. Their persistence in this campaign only highlights the importance of the Special Counsel’s work.”

Mr. Trump’s attorney struck back with a combative tone of his own. He urged Deputy Attorney General Rod Rosenstein to shut down Mr. Mueller’s investigation because it had been “corrupted” by political bias since the beginning.

“I pray that Acting Attorney General Rosenstein will follow the brilliant and courageous example of the FBI Office of Professional Responsibility and Attorney General Jeff Sessions and bring an end to alleged Russia Collusion investigation manufactured by McCabe’s boss James Comey based upon a fraudulent and corrupt Dossier,” the lawyer, John Dowd, said in a statement to news outlets.

Mr. Trump also pointed to the FBI’s failure to notify a surveillance court that it was relying on a dossier funded by Democrats and the Hillary Clinton campaign when it asked to snoop on an adviser tied to the Trump campaign.

House Republicans released a memo detailing the use of the dossier compiled by former British spy Christopher Steele, though the memo also said Mr. Mueller’s investigation is based on additional material.

Republicans on the House Permanent Select Committee on Intelligence said their own investigation hadn’t produced evidence of collusion, prompting Mr. Trump to say Mr. McCabe and other FBI officials had been unfair to him.

“There was tremendous leaking, lying and corruption at the highest levels of the FBI, Justice & State,” Mr. Trump tweeted.

Mr. McCabe reportedly kept memos detailing his interactions with Mr. Trump — materials that Mr. Mueller may use to weigh obstruction of justice charges against the president.

James B. Comey, who was fired as FBI director in May, told Congress that he kept similar memos. Mr. Trump said he doubted either man drafted accurate recollections.

“Spent very little time with Andrew McCabe, but he never took notes when he was with me. I don’t believe he made memos except to help his own agenda, probably at a later date. Same with lying James Comey. Can we call them Fake Memos?” He tweeted.

Some of Mr. Trump’s Republican allies in Congress said the White House should be more cautious.

Sen. Lindsey Graham, South Carolina Republican, said Mr. McCabe’s firing has little to do with Mr. Mueller’s broader investigation.

“I see no cause when it comes to Mr. Mueller,” Mr. Graham told CNN.

Rep. Trey Gowdy, South Carolina Republican and chairman of the House Oversight and Government Reform Committee, said Mr. Dowd was doing the president a disservice by trying to short-circuit Mr. Mueller’s work.

“If you have an innocent client, Mr. Dowd, act like it,” he said on “Fox News Sunday.”

Mr. Gowdy said it’s unclear what, if anything, the House can do if Mr. Trump does fire Mr. Mueller, though the Senate might make life difficult for the president. For instance, it may refuse to confirm a new attorney general if Mr. Trump scraps Mr. Sessions and Justice Department brass as a precursor to reaching Mr. Mueller.

Sen. James Lankford, Oklahoma Republican, said he hopes it doesn’t come to that.

“I don’t see the president firing him,” he told ABC’s “This Week.” “I think the White House has said 10 times, maybe more, that they are not going to fire Robert Mueller.”

• Dave Boyer and Jeff Mordock contributed to this article.

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