WASHINGTON -- The president's rage about an FBI raid on the office and home of his lawyer sparked fresh warnings Tuesday against firing the special counsel Robert Mueller.

Such a move, lawmakers from both parties said, would invite dire consequences -- and could be a tripwire for impeachment. Some Democrats have long itched for that outcome, while most Republicans have insisted that can only be justified by clear presidential meddling and obstruction of justice.

Texas Sen. John Cornyn, the deputy Senate majority leader, said that firing Mueller would be a "big mistake."

"I don't think he or I or anybody could predict what the consequences might be but there would be a lot of backlash, so I think he should just let Mr. Mueller do his job," he said.

Republicans control Congress, though Democrats have a strong shot at taking back the House in November's elections. Impeachment is a two-step process that starts in the House, with trial in the Senate. The appetite for such a step has not existed. But that could change fast if Trump blocks Mueller's inquiry or the investigation of longtime Trump lawyer Michael Cohen.

Cohen is reportedly under investigation for, among other things, a $130,000 hush money payment to porn star Stormy Daniels, who has been trying to sever a nondisclosure agreement about an alleged affair with Trump.

"It could cause some problems if Mueller is fired," Texas Gov. Greg Abbott, a Republican and Trump ally, told Fox Business Network. "I don't know if that would be the smartest move."

Tuesday was chaotic at the White House.

The morning after news broke of the raids on Cohen, Trump scrapped a weekend trip to Peru for a summit with Western Hemisphere leaders. The theme of the summit: "Democratic Governance against Corruption."

The White House said he needed to focus on the U.S. response to a chemical attack in Syria. Soon after that announcement, Trump's well-regarded homeland security adviser, Tom Bossert, resigned -- pushed out, apparently, by the new national security adviser John Bolton.

At the Capitol, reporters bombarded lawmakers with questions about the possibility of Trump firing Mueller or his boss, deputy Attorney General Rod Rosenstein. Many Republicans downplayed the scenario -- but left the clear implication that support for Trump would crumble if he tries to derail the inquiry.

"It would be a disaster for him, and it would be a disaster for a lot of people," said Utah Sen. Orrin Hatch, adding that Trump won't fire Mueller unless he "goes way off track."

"I didn't think that was something they should've done," Hatch said of the raid on Cohen's office. "But I still don't think that's going to cause the president to try to remove him."

Trump's tweets Tuesday morning reflected fury at the direction the investigations have taken.

"A TOTAL WITCH HUNT!!!" he tweeted.

A TOTAL WITCH HUNT!!! — Donald J. Trump (@realDonaldTrump) April 10, 2018

Attorney–client privilege is dead! — Donald J. Trump (@realDonaldTrump) April 10, 2018

"The president has voiced his frustration with the situation," White House press secretary Sarah Huckabee Sanders said.

Democrats want legislation

Democrats renewed demands Tuesday for legislation preemptively stopping Trump from firing the special counsel, appointed initially to investigation allegations of Russian interference in the 2016 election.

Hey @LouDobbs - the American people aren't fooled - the #SpecialCounsel should be permitted to continue his investigation- let's protect the Special Counsel thru legislation that has the support of all @HouseJudDems - pass H.R. 3654 the Special Counsel Independence Protection Act https://t.co/zUVDGPCGym — Sheila Jackson Lee (@JacksonLeeTX18) April 10, 2018

Cornyn and many other Republicans argued that such a measure is unnecessary because Trump won't try to take such a step, a stance echoed by Senate Majority Leader Mitch McConnell, R-Ky.

"Mueller should be allowed to finish his job....I don't think he's going to going to be removed from this office. He shouldn't be removed from this office," he said.

The implications of the raids are significant, though it's not yet known whether federal investigators view Cohen as a gateway to implicating Trump himself.

Strong case

The legal hurdles involved with penetrating the attorney-client privilege suggest the FBI has a strong case.

"Law firms are not evidence free zones. Law degrees do not serve as immunity from prosecution for lawyers," Rep. John Ratcliffe, R-Heath, a former U.S. attorney in Texas, told Fox News. "But I can tell you as a former U.S. attorney, the procedures that you have to go through to even seek a warrant to go into a lawyer's office are layered and significant. They would have had to have approval at Main Justice from the highest levels. My guess is that Rod Rosenstein signed off on this."

On the other hand, he said, the fact that Mueller referred the Cohen investigation to the U.S. attorney's office in Manhattan indicates that it's not directly related to the Russia influence probe at the heart of the special counsel's mandate.

"If this were essential to Bob Mueller's case he wouldn't have made a criminal referral here," Ratcliffe said.

Maine Sen. Susan Collins, a moderate Republican, said that firing Rosenstein "would be an extraordinary crisis and a real problem and I just don't think he's going to do it."

On the Democratic side, Rep. Al Green of Houston, who has long pushed for Trump's impeachment, mocked Trump for trying to deflect blame for his own predicament.

The FBI raid of Michael Cohen's office and home already has Trump fuming. True to form, the Deflector-in-Chief is searching for someone to blame before we even know what was found. It seems that everyone is on the chopping block but the one responsible. #RepealandReplaceTrump — Congressman Al Green (@RepAlGreen) April 10, 2018

Who could fire Mueller

Sanders maintained that the White House believes that Trump has the authority to fire Mueller directly, and to fire Rosenstein.

Many legal experts believe that only Rosenstein has the authority to fire Mueller -- meaning that Trump would have to order him to do so. And Rosenstein would be expected to refuse such an order.

Trump again fumed Monday night at Attorney General Jeff Sessions for recusing himself from the Russia investigation, shifting control of the special counsel to his deputy.

He called the investigation a "disgrace" and said of Mueller: "Many people have said, 'You should fire him.'"

Washington correspondent Katie Leslie contributed to this report.