President Trump Donald John TrumpObama calls on Senate not to fill Ginsburg's vacancy until after election Planned Parenthood: 'The fate of our rights' depends on Ginsburg replacement Progressive group to spend M in ad campaign on Supreme Court vacancy MORE and his allies are pushing back against the deepening probe into allegations of collusion with Russia, but the effectiveness of that strategy is in serious question even among Republicans.

Team Trump has repeatedly suggested that the Russia investigation is a witch hunt, and that special counsel Robert Mueller has conflicts of interest or is exceeding the original bounds of his investigation.

But Republican strategist Ryan Williams argued that Trump’s time would be better spent focusing on his policy agenda rather than voicing his resentment over the purported unfairness of the investigation.

“Let your outside counsel handle the issue, and as president focus on governing,” Williams advised, adding that Trump should act “as if all is well and there is nothing to be concerned about.”

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Instead, he said, “the president continues to obsess about this investigation. That keeps it in the news and it raises questions as to why he seems fixated on this one subject.”

There may not be a cohesive strategy behind Trump's counter-offensive on Mueller and his investigation into potential Trump campaign ties to Russian meddling in the 2016 election, but some conservatives say the White House's defensiveness could pay off eventually.

“Over time it may be beneficial,” said conservative strategist Keith Appell. “Right now I’m not so sure. But it could pay big dividends if Mueller goes over the red line, if he goes too far from his original scope of Russian ‘collusion’ — of which there is no evidence.”

The investigation took another turn for the worse from Trump’s perspective this week, when it emerged that Mueller had begun using a grand jury in Washington.

The move was not entirely unexpected — grand juries are a key weapon in the armory of prosecutors because reticent witnesses can be subpoenaed to appear before them or to produce documents.

But the prospect of people in Trump’s orbit being subpoenaed is a politically grim one for an administration already struggling with historically low approval numbers and a lack of progress on its legislative agenda.

At a rally in West Virginia on Thursday evening, Trump described the allegations as a “total fabrication” and mockingly asked the large crowd whether there were any Russians in attendance.

Earlier at the same event, his daughter-in-law Lara Trump described the Russia “story” as “crazy.”

Strategists like Williams argue that Trump’s apparent inability to let the matter go is counter-productive, especially at events like the Thursday rally, when it was sure to overshadow what was otherwise a positive night for the president.

“Here he is in front of one of the most friendly crowds he is going to face. Yet he can’t help himself but bring up the Russia investigation in a flippant manner,” Williams said. “No-one asked him about it.”

Still, with new developments coming thick and fast, the president cannot magically keep the Mueller probe out of the headlines. Trump has long adopted the stance that attack is the best form of defense in his business career, and he is following that template now.

The specific lines of attack against Mueller met with disapproval from one former Department of Justice official who spoke to The Hill.

“I think it is an exceptionally weak argument. Bob Mueller to anyone who knows him is Mr. Fairness with a capital ‘F’,” said Harry Litman, a former U.S. attorney and deputy assistant attorney general under former President Bill Clinton William (Bill) Jefferson ClintonBarr says Ginsburg 'leaves a towering legacy' Trump reacts to Ginsburg's death: 'An amazing woman who led an amazing life' Jimmy Carter remembers Ruth Bader Ginsburg as 'a beacon of justice' MORE. “All indications are that this is a professional — if aggressive — and unbiased investigation."

But Trump has persistently argued that the investigation is specifically out to get him.

Asked in a New York Times interview in mid-July whether Mueller would be crossing red lines if he began to investigate Trump’s financial affairs beyond Russia, the president replied, “I would say yes.”

He added that any such move would be “a violation. Look, this is about Russia.”

Mueller's investigation is now reportedly looking into Trump family business ties.

Meanwhile, Trump allies are also spoiling for a fight.

Pro-Trump outside groups are fundraising off the issue, warning supporters that the “witch hunt against President Trump has reached a fever pitch.”

“The only confirmed collusion is that between the Washington establishment and the independent ‘investigators’ who are using government resources and taxpayer funds to perpetuate this witch hunt,” Eric Beach, the head of the pro-Trump Great America Alliance, wrote in a fundraising email to donors.

Ron Hosko, former assistant director of the FBI’s criminal investigation division who worked under Mueller, said the special counsel would not be immune to public pressure to finish the investigation but said that he is unlikely to change course because of it.

“He’s not going to be deaf and blind to those pressures, but he’s going to see them for what they are,” Hosko said.

“With Bob Mueller, there’s always a sense of urgency,” he added.

Trump allies who may be called before the grand jury will be obligated to tell the truth or face serious legal jeopardy. Those hard and fast rules are very different from the "anything goes" style of political rhetoric.

“A notice to appear in front of a grand jury is a very sobering event,” said Litman. “There are going to be people within Trump’s orbit who, in the next couple of months, are going to be called before either investigators or a grand jury. That is going to change the dynamic not just of the investigation but within the White House itself.”

One member of the president’s outside legal team emphasized this week that Trump has no intention of firing Muller, after a period in which rumors had swirled to that effect.

Jay Sekulow told Neil Cavuto of Fox News, “The president is not thinking about firing Bob Mueller, so the speculation that’s out there is just incorrect.”

Still, with Trump now on a working vacation at his New Jersey golf club, he will have plenty of time to stew over the probe. What comes next is anyone’s guess.

The Memo is a reported column by Niall Stanage, primarily focused on Donald Trump’s presidency.

Jonathan Easley and Morgan Chalfant contributed to this report.