Analysts say that reaction could be short-lived, though it may herald a period of uncertainty and volatility for markets.
"Obviously, the knee-jerk reaction is to sell because of the uncertainty that would ensue, but in terms of really affecting the U.S. economy, interest rates and earnings, it's not going to mean anything. The presidency has been a constant drama ... while this might take it to a different level, I'm not sure what the relevance would be to the market," said Peter Boockvar, chief investment strategist at Bleakley Financial Group.
The White House said Tuesday that Trump "certainly believes he has the power" to fire Mueller and that the Michael Cohen investigation has "gone too far."
A more prolonged period of market turbulence could come if there's an effort to impeach Trump, but Daniel Clifton, head of policy research at Strategas, said that could backfire on Democrats as it did on Republicans when they voted to impeach President Bill Clinton.
One of the market's longer-term concerns is for a big Democratic victory in the congressional midterm elections this fall. Clifton said in 1998, impeachment was a big theme during the midterms, but the more the GOP's impeachment efforts moved forward, the more Democratic voters were energized.
"The Republicans had a very wide lead in the midterm election in the beginning of 1998. As Clinton ran into problems in July and August, the Republicans started raising impeachment. What was interesting was as Clinton runs into these problems, you see the polling data improve for the Democrats," said Clifton.
But still, the period around the 1998 impeachment effort was rocky for stocks. which were also impacted by other factors like the Asian flu, Clifton said.
"You can literally take the developments of the special counsel and overlay them with S&P 500," Clifton noted. In October 1998, the House initiated an impeachment inquiry and Clinton was impeached by the House in December 1998 on charges of lying under oath and obstruction. But he was acquitted by the Senate in February.
Source: Strategas Research
Strategists said the market's response to news Monday that special counsel Mueller had initiated an FBI search of Trump's personal lawyer's office and residence sent stocks reeling and the dollar, already lower, remained under pressure.
"What we saw yesterday was a test case," said Marc Chandler, head of foreign exchange strategy at Brown Brothers Harriman. "The knee-jerk reaction would be to sell the dollar. But I say that because we sell the dollar for no reason."
After news broke about the FBI raids Monday afternoon, Trump railed against Mueller, saying his investigation was a "witch hunt" and "an attack on our country in a true sense." The FBI reportedly sought payment records related to porn star model Stormy Daniels and a Playboy model, who both said they had affairs with Trump.
Trump also repeated that he would never have appointed Jeff Sessions as U.S. attorney general if he had known he would recuse himself in the Russia investigation.
Strategists said it's possible Trump could fire Mueller but more likely he will fire his bosses and hire someone who would narrow the scope of Mueller's investigation to the topic of collusion with Russia on election tampering. Trump has repeatedly said there was no collusion between his campaign and Russia.
"I suspect he won't get fired. I suspect people above him will get fired first, to change his mandate," said Chandler.
Clifton said he would expect that Rod Rosenstein, deputy attorney general, could be the one to get fired, if Trump dismisses anyone. Rosenstein reportedly signed off on the raids.
"Our base case has been a recommendation of obstruction of justice. You can't impeach Trump on obstruction of justice. They'd have to get him on collusion," said Clifton.
Strategists say even if Trump were charged with something and there efforts to remove him, the market could react to the uncertainty but would be comfortable with Vice President Mike Pence.
Clifton said if that were to happen it would be a 2019 time frame. "The market would look through it to Pence. ... It's a bigger problem if your economy is tanking," he said.
Boockvar said the market has gotten what it expected from Trump — a major corporate tax cut.
"From a legislation standpoint, Trump pretty much got what he wanted with the cuts in the corporate income tax rate. I don't think there's anything else on his agenda that if it doesn't happen, it would be a big deal. The trade stuff will go on its own path regardless of what happens with Mueller," said Boockvar.