Will he, or won't he?

The question of whether Donald Trump will fire the special counsel investigating Russian interference in the 2016 election hovers over both ends of Pennsylvania Avenue, as the president recently lashed out at Robert Mueller by name and added a new and aggressive attorney to his personal legal team.

But even if he doesn't decide to shut down the probe -- action that Sen. Lindsey Graham and others have said would amount to a "constitutional crisis" -- Trump could still be getting some of what he wants in terms of public perception.

For starters, voter sentiment about the Mueller investigation, which includes scrutiny of possible collusion with the Russians by Trump associates and potential obstruction of justice by the president, has become increasingly partisan. A new NBC News/Wall Street Journal survey finds that 33 percent of Republicans hold a negative view of Mueller, constituting a new high for the poll. And just 11 percent held positive feelings. Meanwhile, 47 percent of Democrats held positive feelings about Mueller, also a new high for the poll.

A Pew survey released late last week found similar divisions, with 46 percent of Republicans confident in Mueller's ability to conduct a fair investigation, while 75 percent of Democrats were confident. Moreover, the survey found 75 percent of Republicans say Russian meddling didn't benefit either campaign, while an equal number of Democrats say it benefited Trump.

"This is a campaign, just like a candidate's campaign or initiative. ... Trump is engaged in a long-term persuasion campaign among the American people to undermine the credibility of those conducting the investigation," says Republican strategist Rick Tyler.

While it's unclear whether negative perceptions of Mueller among the GOP electorate will continue to increase over the coming months, Trump has been testing the waters to how far he might be able to go with his party’s lawmakers, who still hold the majority in Congress. Most have been urging him to simply let the investigation play out, but they are also reluctant to move legislation to protect the special counsel. "Obviously, legislation requires a presidential signature," Texas Sen. John Cornyn told reporters this week. "I don't see the necessity of picking that fight right now."

And some GOP strategists say they don't see much evidence of the probe having an impact on Republican voters at this point."Once Mueller is finished with his work, voters will judge the full findings and the circumstances of them and make their decision," says party operative Brad Todd. "Until then I think voters are worried about their paychecks and their retirement accounts."

GOP leaders on Capitol Hill insisted Tuesday they are confident Trump won't end the probe. House Speaker Paul Ryan has said he wants Mueller to complete his work undeterred and told reporters he has received "assurances" that the president isn't going to fire the special counsel.

Senate Majority Leader Mitch McConnell finally weighed in on Tuesday, expressing similar sentiments. "Bob Mueller should be allowed to finish his job. I think it was an excellent appointment. I think he will go wherever the facts lead him and I think he will have great credibility with the American people when he reaches the conclusion of this investigation," McConnell said. "This is a thoroughly credible individual, I think it was an appropriate appointment, and we all anticipate his finishing the job and telling the American people what they need to know about this episode."

But Trump doesn't show any signs of letting up on his campaign against Mueller. This week, he hired attorney Joseph diGenova, who has been outspoken in promoting conspiracy theories about the Russia investigation. The hiring signals the president is ready to take a more combative approach going forward.

White House spokeswoman Sarah Sanders suggested that Trump will continue to air his frustrations.

"The president has been very clear about the fact that there was no collusion between his campaign and any other entity. However, to pretend like going through this absurd process for over a year would not bring frustration seems a little bit ridiculous," Sanders told reporters Tuesday.

Sanders then asked whether members of Congress "would like it if they had been accused of taking their seat in Congress by doing something nefarious when they hadn't, particularly if it went on for more than a year into their time in office? My guess is they would be more than anxious to push back."

Though Republican lawmakers may sympathize with the president, there is also frustration over the way Trump continues to make the probe a story. "When you're innocent, act like it!" South Carolina Rep. Trey Gowdy, a former prosecutor and chairman of the House Oversight Committee, said during an interview with Fox News. "If you’ve done nothing wrong, you should want the investigation to be as fulsome and thorough as possible."

Gowdy's South Carolina colleague, Graham, warned Trump that firing Mueller would likely be an impeachable offense. "Probably so, if he did it without cause, yeah," he said on Hugh Hewitt's radio program. "What the president will have done is stopped an investigation into whether or not his campaign colluded with the Russians, what effect the Russians had on the 2016 campaign. I can’t see it being anything other than a corrupt purpose."

Retiring Arizona Sen. Jeff Flake, a frequent critic of the president who has spurred talk of him challenging Trump in 2020, said Republicans want Mueller left alone. "Congress cannot preempt such a firing. Our only constitutional remedy is after the fact, through impeachment. No one wants that outcome," Flake tweeted. "Mr. President, please don't go there."

Even though Flake is seen as an outlier within the party, there has long been frustration among Republicans up for re-election this year that the president continues to step on the GOP's campaign message, particularly its economic accomplishments.

"The success of the Republican Party isn't the foremost thing on the president's mind. It's ‘How do I fare and look in all of this?’ " Tyler says.

Michael Steel, a GOP strategist and Capitol Hill veteran, says Republicans aren't under any illusions that the president will change course. "Most Republicans would be happier if the president would spend every day talking about the benefits of the tax cuts ... but I don't think anyone has a realistic expectation he's going to do that, so it's then incumbent on candidates" to make that case, he says.

"However Republicans feel about the president and the investigation, it's critically important for Republican voters to turn out to vote for Republicans. ... A Democratic majority in either chamber would block any further efforts to improve our economy and be completely obsessed and bogged down with investigations of the administration and ultimately, most likely, impeachment proceedings."