WASHINGTON – President Trump has publicly called the widening federal investigation into Russia's election meddling a "witch hunt." But through his lawyer, Trump has sent private messages of "appreciation" to special counsel Robert Mueller.

"He appreciates what Bob Mueller is doing,'' Trump's chief counsel John Dowd told USA TODAY in an interview Tuesday. "He asked me to share that with him and that's what I've done.''

Trump's legal team has been in contact with Mueller's office, and Dowd says he has passed along the president's messages expressing “appreciation and greetings’’ to the special counsel.

“The president has sent messages back and forth,’’ Dowd said, declining to elaborate further.

Washington political circles for weeks have been on high alert for the possibility the president could try to get rid of Mueller or otherwise wrest control of the federal Russia probe he frequently denounces as a "hoax."

Yet the Trump legal camp cast the communications as a sign that the president is willing to cooperate with Mueller, whose office declined to comment, and former federal prosecutors say there are no rules that would prevent a president from passing along messages to the special counsel through an attorney.

Even so, Trump's private communications represent a "striking" contradiction with his public condemnations of the Russia investigation, said Jimmy Gurule, a former assistant attorney general in the George H.W. Bush administration.

"It's an extreme position to go from hateful statements to this," Gurule said. "What would cause him to change his mind?"

Trump's effort to communicate with Mueller is also notable since Trump has come under fire for prior unconventional contacts related to the investigation into possible collusion between Trump associates and Russia.

In a break with longstanding precedent to avoid even the appearance of influencing the nonpolitical law enforcement agency's investigations, Trump spoke privately with former FBI Director James Comey on several occasions before firing him in May because of his handling of the Russia matter.

In those conversations, Comey said, Trump tried to convince him to drop parts of the Russia investigation and asked for a pledge of loyalty – accounts that not only led to the appointment of a special counsel, but also an expansion of the inquiry to include possible obstruction of justice.

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Dowd has said all communications with Mueller have been proper.

“We get along well with Bob Mueller; our communications have been constructive,’’ the attorney said. “But it is important that our communications remain confidential. It’s important that there not be any breakdown in that trust.’’

In a sign that forcing out the former FBI director overseeing the federal Russia investigation is also untenable from a political perspective, government officials have moved to tamp down any talk of ousting Mueller – including, apparently Trump himself.

"That’s never been on the table, never,’’ Dowd said of the possibility Trump might try to remove the special counsel. “It’s a manifestation of the media. My dealings with Bob Mueller have always been cordial, respectful — the way it should be."

Ron Woods, a former federal prosecutor and FBI agent, said he saw nothing inappropriate about Trump’s communications to Mueller through his attorney.

"I think it’s a smart play on Dowd’s part," Woods said. “They’ve obviously taken the position that the best defense now is to cooperate."

While Woods notes that "nothing about this case has been usual," both Dowd and Mueller have a history as friends. "There is nothing wrong with Dowd talking to Mueller," Woods said.

Not all legal experts agreed it was so straightforward.

"This is peculiar," tweeted Preet Bharara, a former U.S. attorney who was fired by Trump earlier this year. "It would be more helpful if he stopped undermining Mueller publicly and fomenting attacks on him by surrogates like Newt (Gingrich)."

Trump and his aides have been highly critical of Mueller and his widening investigation. Trump has said Mueller's job is unnecessary because he hasn't done anything wrong, and has accused the former FBI director of having unspecified conflicts of interest.

Lawmakers, including Republicans, started sounding the alarm about Mueller after days of concerted attacks on Attorney General Jeff Sessions. The unusual public criticism of a cabinet official who was one of Trump's earliest supporters – months after his recusal from the Russia investigation – raised questions about whether the president was trying to get Sessions to quit or would fire him.

Trump cannot fire Mueller directly, but he can fire Sessions. And a new, more pliable attorney general – without conflicts in the Russia investigation – could oust the special counsel.

Yet the two men appear to be at a stalemate: Sessions has has refused to resign in the face of attacks by Trump, and so far, Trump has not moved to fire Sessions.

And senators from both parties have since introduced legislation that would make it harder for the president to dismiss Mueller. A pair of pending proposals would require judges to to review any presidential firing, and force the president to provide specific legal reasons for taking such action.

What's more, with the public now aware that Mueller has tapped a grand jury for his inquiry into possible collusion between Trump associates and Russians who sought to influence the election, getting rid of the top prosecutor seems like a near impossibility.

One Republican senator, Lindsey Graham of South Carolina, said late last month any move against Mueller could be "the beginning of the end of the Trump presidency."

Sen. Thom Tillis, a North Carolina Republican who is sponsoring one of the proposals designed to protect a special counsel, said that "a back-end judicial review process to prevent unmerited removals of special counsels not only helps to ensure their investigatory independence, but also reaffirms our nation’s system of checks and balances.”

A recent poll of battleground districts indicated that firing Mueller would be unpopular with voters as well.

Mueller is supervising a long-running inquiry that focuses on Russian efforts to influence the 2016 election through fake news and hacking political organizations and Democrats close to nominee Hillary Clinton.

The wide-ranging inquiry now includes a June 2016 meeting at Trump Tower where the president’s eldest son, Donald Jr., hosted a Kremlin-linked attorney whom he believed had damaging information on Clinton.

Trump Jr. was told that that information would be provided by the Russian government. Trump's son-in-law Jared Kushner and then-campaign chairman Paul Manafort also attended the meeting.

Trump Jr. and the Russian attorney both said no actual information about Clinton was provided at the meeting. The special counsel’s team also is reviewing financial transactions involving campaign officials.

As the probe continues to mount, Trump and his aides have complained that some of Mueller's staff members have given political contributions to Democrats. Trump has also objected to the idea of having Mueller investigate his personal finances.

"No, I think that’s a violation," he told The New York Times. "Look, this is about Russia."

But this has not deterred Mueller, who is now using at least two grand juries — in Virginia and Washington, D.C. — to advance the investigation.

The use of the grand jury located in Washington, first disclosed last week, is potentially significant, as it likely means investigators are examining activities that happened within that jurisdiction. Those actions include former National Security Adviser Michael Flynn’s pre-inaugural contacts with Russian ambassador Sergey Kislyak.

Shortly after his appointment in May, Mueller also took control of an existing federal grand jury in Alexandria, Va., that had been examining Flynn's prior lobbying business involving Turkey.

While Trump and some of his aides have criticized Mueller and his operation, they also said there is no sign that the president himself is under investigation.

Dowd said he knew of no immediate request for Trump’s testimony, and he declined to comment on any request for documents related to the president.

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