Lindsey Graham, Rosenstein said that having given political donations is not a disqualifier.
The public testimony Tuesday before the Senate Intelligence committee should yield Sessions' most extensive comments to date on questions that have dogged his tenure as attorney general and that led him three months ago to recuse himself from the Russian Federation probe.
Until now, Mueller had drawn widespread praise from Republicans and Democrats alike.
US President Donald Trump is considering terminating special counsel Robert Mueller, who was appointed to lead the FBI investigation into Russia's alleged ties to the 2016 US presidential election.
House Speaker Paul Ryan told reporters that any talk about Mr Trump wanting to rid himself of the special investigator amounts to rumour.
"I regret that I do not", Rosenstein said. "If there were not good cause it wouldn't matter what anyone said". But he told The Palm Beach Post his comment that Trump was "considering" firing Mueller was based on conversations with "a half dozen people familiar w pres thinking on the matter".
Attorney General Jeff Sessions, whose contacts with Russia's ambassador to the US during the presidential campaign have sparked questions, is appearing before the Senate intelligence committee as it investigates alleged Russian meddling in the election.
When asked to clarify the boundaries of Attorney General Sessions' recusal from the Russian Federation probe, Rosenstein said, "I know what we're investigating and (Sessions) does not".
Rosenstein said "no, I have not", when asked by Sen. Comey said he and other Federal Bureau of Investigation officials expected Sessions to recuse himself before it was publicly announced but would not discuss why they came to that conclusion.
Rod Rosenstein told lawmakers of a Senate appropriations subcommittee that he has seen no good cause to remove special counsel Robert Mueller from the probe.
Trump Warns Travel Ban Ruling Comes at a 'Dangerous Time'
Trump's first order on January 27 led to chaos and protests at airports and in various cities before being blocked by the courts. This is seen as a major defeat for Trump's proposed temporary ban for travelers from six predominantly Muslim countries.
The headline-making question from the Senate hearing was posed by New Hampshire Democratic Sen. Lindsey Graham: "Do you know of any reason for cause to fire Mr. Mueller as of this date?" After responding that he had not, Rosenstein reassured Shaheen of his confidence in the process.
"The attorney general could fire him, that's the only person who could fire him."
Democrats, reacting to the chatter, said that if Mr Trump fired Mr Mueller they would try to enact an independent counsel statute so they could appoint him.
Christopher Ruddy, chief executive of Newsmax Media, had been at the White House prior to his conversation with PBS NewsHour's Judy Woodruff, the New York Times reported. There is increasing concern at the White House and among Trump supporters that the investigation could overshadow the president's agenda for months to come. "That, to me, is the smartest thing to do, the best thing to do".
"I don't think Trump should do anything but the congressional Republicans ought to look into it", Gingrich said.
The afternoon p.m. hearing will be the first time that Sessions is questioned by lawmakers since January when he testified during his confirmation hearing that he did not communicate with any Russian officials during the presidential campaign, when Sessions acted as an adviser to Trump.
But he could order the Justice Department to do so.
Leahy says he wants to know why Sessions "has provided false testimony" about those contacts.
Legal experts say the president still has authority to fire Mueller, who was appointed to investigate Russia's meddling in last year's presidential election, even though the special-counsel statute that existed when the Watergate cover-up was being investigated was changed after the long investigation of the Clinton presidency.
The White House forcefully disputed Ruddy's claims, saying he speaks for himself, not the president.
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