In May, Rosenstein appointed former FBI Director Mueller as special counsel overseeing the investigation into Russian interference in the 2016 election.
Trump's allies have also begun raising questions about special counsel Robert Mueller's impartiality - he's a former FBI director who has worked with fired FBI Director James Comey - and floating the idea that Trump might replace him.
"While the president has every right to" terminate Mueller, Sanders told reporters, "he has no intention to do so".
Mueller has been silent since he was named to head the Russian Federation investigation on May 17, a week after Trump fired FBI chief James Comey expressing frustration over the probe. Susan Collins, R-Maine, asked Rosenstein what he would do if the president asked him to fire Mueller.
"It's a mistake to pretend that this is going to be some neutral investigation", Gingrich said.
Later Tuesday afternoon, Democratic lawmakers intend to ask Attorney General Jeff Sessions about his meetings with the Russian ambassador and the firing of former Federal Bureau of Investigation director James Comey when the former Alabama senator testifies Tuesday.
"And the reason for that is, that if it is within the scope of Director Mueller's investigation - and I've been a prosecutor for 27 years - we don't want people talking publicly about the subjects of ongoing investigation", he continued. According to someone who spoke to Trump, "The president was pleased by the ambiguity of his position on Mr. Mueller, and thinks the possibility of being fired will focus the veteran prosecutor on delivering what the president desires most: a blanket public exoneration". This comes after a friend of the President says he thinks Mr. Trump is considering removing Mueller. Now, however, some Republicans are suggesting that Mueller ought to be dismissed, pointing to Mueller's selection of key investigators who made campaign donations to Trump's Democratic challenger, former U.S. Secretary of State Hillary Clinton.More news: Trump confirms he is being investigated over Comey firing in tweet
Ruddy appeared to be basing his remarks, at least in part, on comments from Jay Sekulow, a member of Trump's legal team, who told ABC in an interview Sunday that he was "not going to speculate" on whether Trump might at some point order Rosenstein to fire Mueller. I know Bob Mueller. Ruddy was asked again later in the interview if Trump was actually contemplating firing Mueller, to which he replied, "Yes".
Gingrich also said Trump is confident that Mueller will find he did nothing wrong, but Gingrich worries.
President Donald Trump speaks to Associated Press Chief White House Correspondent Julie Pace in the Oval Office in Washington, Wednesday, April 19, 2017. "And the next day, he's now maybe using some of that information in his investigation". "That, to me, is the smartest thing to do, and that's what I think hopefully will happen".
MARGARET BRENNAN: Republican leaders Mitch McConnell and Paul Ryan expressed support today for Robert Mueller, the special counsel investigating Russian election meddling and allegations of Trump campaign collusion.
"I'm not saying this investigation isn't important", said Ryan.
The Justice Department declined CBS News' request for comment.
As attorney general, Sessions is unlikely to answer in detail questions about conversations he's had with Trump.