"It would be a catastrophic mistake, but he doesn't have the authority to do it", Republican senator Susan Collins of ME told reporters.
When Mueller was appointed as Special Counsel, Newt Gingrich praised the selection.
A spokesman for Mr Mueller declined to comment.
USA TODAY previously reported that the FBI was investigating Kushner for his contact with Russian officials, but there was no word at that time that his financial dealings were part of the probe.
Mueller is also investigating whether President Donald Trump attempted to obstruct justice, The Washington Post reported Wednesday.
Officials familiar with the matter describe friction on the Justice Department's fourth and fifth floors, home to the suite of offices belonging to the deputy attorney general and the attorney general, respectively, in part because of Rosenstein's handling of the Russian Federation matter. Jay Sekulow, a member of the president's personal legal team, told CNN's State of the Union on Sunday that it was in response to the Post report.
The New York Times reported that former members of Trump's transition team were given "preservation orders" Thursday to safeguard documents, including electronic records like email, related to the ongoing Russian Federation investigation.
Mr Comey told Congress last week he believed he was sacked by Mr Trump to undermine the agency's Russian Federation probe. "He will do the same if he is contacted in connection with any other inquiry". Then they thought at least they could deflect questions to Trump (who in turn would direct questions to his high-octane lawyer Marc Kasowitz) or to the special counsel. He is reportedly interviewing three top intelligence officials - Director of National Intelligence Dan Coats, National Security Agency Director Mike Rogers and retired deputy NSA director Richard Ledgett - as part of the probe as early as this week.
The former director of the Department of Homeland Security, Jeh Johnson, is scheduled to appear June 21 in an open session before the House of Representatives' intelligence committee, which is looking into Russian activity during the 2016 US election campaign. Although Trump polled at an average of 37 percent job approval for the week ending June 11, the weekly average was the lowest of his administration.
Several news reports said Mueller plans to interview key US national security officials about Trump's comments seeking an end to the investigation of contacts his first national security adviser, Michael Flynn, had with Russia's ambassador to Washington, Sergey Kislyak. But by including his name in the list, Trump attorneys suggest Gates is being or might be scrutinized in relation to the investigations. Mueller's decision to talk with the two officials was reported earlier by the Washington Post. Cohen has received a subpoena from one of the congressional committees looking into the Russian Federation issue.
If Trump and other senior White House officials have concerns about Rosenstein's credibility, they certainly didn't voice it in the past.
"To the best of my recollection, I have never been directed to do anything I believe to be illegal, immoral, unethical or inappropriate", Rogers said at the hearing, without answering whether he was asked - but not directed - to back off.
Flynn, who was forced out in February, is a subject in investigations by intelligence committees in the House and Senate, as well as the Federal Bureau of Investigation. Flynn was forced to resign for misleading administration officials about his contacts with Russia's US ambassador. Under the specific circumstances, however, with his longtime friend James Comey at the center of the inquiry, Mueller's the wrong choice.
Rosenstein took over the Russian Federation probe soon after Attorney General Jeff Sessions recused himself.
Dianne Feinstein, top Democrat on the Senate Judiciary Committee, said she was "increasingly concerned" that Trump will fire both Mueller and Rosenstein.
"It's frustrating because it keeps us from getting to the things that need to be done, and I'm not sure that that isn't the real reason for it", he said.
Challenged on "Fox News Sunday" about the issue, Sekulow said he could not be certain Trump was not under investigation but that no one had notified the legal team if that was the case.