"The special counsel is subject to the rules and regulations of the Department of Justice, and we don't engage in fishing expeditions," Rosenstein said when asked about the probe in an interview on "Fox News Sunday."
Rosenstein declined to comment on reports that Mueller is using a grand jury in a court in Washington to aid in his investigation but he said that such a step is a routine part of "many investigations."
"It's an appropriate way to gather documents, sometimes to bring witnesses in, to make sure that you get their full testimony," Rosenstein said. "It's just a tool that we use like any other tool in the course of our investigations. "
Trump and his inner circle have repeatedly dismissed the investigation amid frequent reports that Mueller and his team are digging into broader details on the financial dealings of members of Trump's campaign team. Senior White House counselor Kellyanne Conway called the probe a "fabrication" in an interview on ABC's "This Week." Trump called it "the totally made-up Russia story" in a campaign-style speech he delivered Thursday in West Virginia.
The attacks have raised concerns among Democrats and some Republicans that Trump may be looking for ways to undermine the investigation. Those fears led Sens. Thom Tillis (R-N.C.) and Christopher A. Coons (D-Del.) to propose legislation that would give a judge the ability to review any decision by the president to fire Muller.
Tillis said Sunday that he does not agree that the investigation is a witch hunt and said the bill is intended to bolster the independence of the Justice Department.
"We'll let the facts lead us to whether or not it was a hoax or a distraction," Tillis said during a "This Week" interview. "But we are where we are, and I want to see this investigation concluded so that we can get on to doing the good work the president has already started with regulatory reform, health care and tax reform."
Rep. Adam B. Schiff (Calif.), the ranking Democrat on the House Intelligence Committee, called Mueller's impaneling of a D.C. grand jury "a significant development," noting that it has been more than a year since former FBI director James B. Comey launched a counterintelligence investigation into Russia's meddling in the 2016 election.
"That means one year later, rather than turning that investigation off, rather than concluding 'We've looked at this for a year; there's really nothing to see here,' as the president would claim, instead . . . it's moving into a new phase," Schiff said during an appearance on CNN's "State of the Union." "That wouldn't be taking place if there was really no evidence, no evidentiary basis to move forward."
He said an additional reason to continue investigating was the disclosure of the June 2016 meeting of Donald Trump Jr., campaign officials and a Russian lawyer, which was set up with the advertised purpose of sharing damaging information on Democratic presidential nominee Hillary Clinton.
"And now you add on the layer of the president, if these allegations are true, helping to fabricate a false statement about what that meeting was about," Schiff said, referring to the White House's acknowledgment that Trump weighed in on an initial statement issued by Trump Jr. about the meeting that did not mention its pretext.
Schiff also said the House Intelligence Committee and Mueller are looking at some of the same issues related to former national security adviser Michael Flynn, including payments Flynn allegedly received from Turkey during the final months of the presidential campaign and from RT, a Russian government-backed television network.
"If General Flynn was shown to have violated the law in other ways, it would be an incentive for him to cooperate more broadly with the Mueller investigation," Schiff said.
During an appearance on the same CNN program, New Jersey Gov. Chris Christie (R), an ally of the Trump administration, downplayed the significance of a D.C. grand jury being impaneled by Mueller.
"That's a typical thing to be done in any investigation," said Christie, a former federal prosecutor.
Asked about Trump's concerns that Mueller's probe could expand into financial dealings of Trump unrelated to Russia, Christie said that some sometimes special counsels feel "the need to produce something in return for their appointment."
But he called Mueller "a good man" and said he trusts he will not go on a "fishing expedition."
Christie also called Trump Jr.'s meeting with the Russian lawyer "ill-advised." The meeting was also attended by Jared Kushner, the president's son-in-law and now senior adviser, and Paul Manafort, then Trump's campaign chairman.
"This is not something that should have happened," Christie said. "Everybody in retrospect knows that this is a bad idea."
But Christie said it remains unclear whether Trump was aware of the email to Trump Jr. ahead of the meeting that said it would be related to Russian information on Clinton.
"We don't know that the president knew about those emails or about the content of those emails," Christie said. "And so we don't know what his own son told him about that meeting."
On ABC, Conway said Trump "had no knowledge of that meeting."
"I was never informed of that meeting," she said. "I found out about it when you found out about it, when the rest of the public did."
Conway also referenced Kushner's contention that he left the meeting a few minutes in when it became clear the Russian lawyer wanted to talk about adoption policy. Kushner has said he had an aide pull him out.
"If you're getting dirt on your political opponent, if you're getting the silver bullet and the secret sauce on how to win the election, you don't ask your aide to pull you out of the meeting," Conway said. "You say please order lunch. Let's just stay."