Get breaking news alerts and special reports. The news and stories that matter, delivered weekday mornings.

Congressional Republicans Tuesday pushed back on calls for an independent probe into the chain of events that led to Michael Flynn's stunning late-night resignation as National Security Adviser.

Flynn's departure Monday night followed revelations that he had misled senior administration officials, including Vice President Mike Pence, about the substance of his pre-inauguration conversations with Russia's ambassador to the U.S., Sergey Kislyak, and that the Justice Department had informed the White House that he could be subject to blackmail from Russia.

Congressional Democrats want a broader probe into the matter, while Republicans are insisting that the Senate is already equipped to examine the complete account of his departure.

"We have standing committees in the Senate that have all the appropriate clearances to do the investigations," said Sen. John Cornyn, R-Texas, the second-highest ranking Republican in the Senate. "We'll follow the investigation wherever it leads."

Both Democrats and Republicans in Congress are demanding more answers and a more complete account about what the White House knew about those conversations and when — but the scope and breadth of their desired inquiries differ.

The FBI and the Senate Intelligence Committee had already opened separate investigations into Russia’s interference in the U.S. election — and the Senate investigation could become much broader, members say, to include the circumstances surrounding Flynn.

Sen. Roy Blunt, R-Missouri, a member of leadership and of the Intelligence Committee said "we’re well on the way to looking at those issues and I think this will easily fit into that investigation."

Related: Flynn Resigns as National Security Adviser Over Talks With Russia

"I'd like to know, did he just do this as a rogue, General Flynn just decided to call the Russians up one day and say we're gonna have a different view on sanctions don't worry about it, or did it come from somebody else in the White House?" asked Sen. Lindsey Graham, R-S.C.

Democrats Call for Independent Investigation

Democrats were on the offensive Tuesday as both House Minority Leader Nancy Pelosi and Senate Minority Leader Chuck Schumer called for an independent investigation that would have authority to prosecute criminal activity to look into the issue.

“The truth and consequences of the Russia connection: the American people deserve to know the full extent of Russia's financial, personal and political grip on President Trump and what that means for our national security,” Pelosi said in a statement.

And Schumer said that any investigation conducted by the FBI that is coordinated with the Department of Justice should not include its new attorney general, Jeff Sessions.

Let our news meet your inbox. The news and stories that matters, delivered weekday mornings. This site is protected by recaptcha

"I expect Attorney General Sessions to recuse himself and make sure that independent thorough investigation proceeds," Schumer said, pointing to Justice Department regulations that says an employee cannot have a personal relationship with the person or entity it is investigating.

After intelligence officials revealed Russian interference in the U.S. elections in December, Republican leaders of the House and the Senate agreed to allow respective committees to investigate, but they dismissed calls by Democrats and a few Republicans for an independent outside committee to be formed to conduct the oversight. Similar calls are now being revived.

Republicans Say the Congressional Oversight is Enough

Vice President Pence joined Senate Republicans for lunch Tuesday on Capitol Hill where he briefly walked through what happened with Flynn, Republican senators said.

But calls remain among the GOP for a more robust and independent investigation have not been heeded by leaders of a Republican-controlled Congress that is reluctant to investigate its president and administration.

Senate Majority Leader Mitch McConnell told MSNBC's Joe Scarborough in an interview to air Wednesday that there should not be a select or independent committee.

"We know how to do our work. We have an Intelligence Committee. Over on the Judiciary Committee, (Sen.) Lindsey Graham has a subcommittee that’s gonna take a look at it. I don’t think we need to set up a special committee," McConnell said. "But we are looking into Russian involvement in the U.S. election. That’s a significant issue. We know there was messing around."

House Speaker Paul Ryan would only say that President Donald Trump made the right decision by asking for the resignation.

“You cannot have a national security adviser misleading the vice president and others. So I think the president was right to ask for his resignation. And I believe it was the right thing to do,” Ryan said.

Republicans indicated that the Senate is well-equipped to ask questions without having to set up a separate entity with its own budget and staff to solely focus on the issue.

“There needs to be a full investigation on all angles relative to nefarious activities that were taking place with Russia beginning in March but even going back before that time," said Sen. Bob Corker, R-Tennesee, and chairman of the Senate Armed Services Committee.

Florida Sen. Marco Rubio added: “I can tell you that the Senate Intelligence Committee is conducting a bipartisan investigation, as we should, on Russian interference on our election and post-election and we’re going to go wherever the truth leads us."

Related: Flynn’s Departure Raises More Questions Than It Answers

Rep. Jason Chaffetz, chair of the House Oversight and Government Reform Committee said his committee won't look into Flynn, telling reporters that the Intelligence Committees are better suited.

"That situation has taken care of itself - (the) Intel committee has looked at hacking issue previously," Chaffetz said.

Frustration Growing

Frustration with the administration continues, especially among Republican lawmakers who want to focus on legislation but are blindsided by executive decisions or are forced to respond and focus their communications strategy on responding to the White House on administrative and ethics problems.

Sen. John McCain, R-Arizona, and head of the Senate Armed Services Committee, said in a statement that Flynn’s resignation is “a troubling indication of dysfunction” for the administration.

"We need to look at the whole issue … our relationship with Russia and how certain things happened the way they did," McCain told reporters.

And Sen. John Thune, R-S.D., said that he expects the administration to answer their questions.

"I think it’s important air it out and I think it’s happening," Thune said. "But in the meantime I want to keep us focused."

When asked if he had any indication of the problems surrounding Flynn, Blunt, a member of the intelligence committee, said, “I certainly wasn’t kept informed.”

Concerns over the administration's ties to Russia had previously existed. Calls have emerged for the transcripts of Flynn's phone conversations with the Russian ambassador Kislyak before Trump was inaugurated to be released. Corker said, “I don’t know why that would be harmful."

Some key Republicans shied away from any criticisms, however, and said that the future of the National Security Council is more important than the past several weeks.

House Intelligence Committee chairman Devin Nunes said in a short statement that Flynn served “with distinction.”

“The President needs a National Security Advisor whom he can trust and I defer to him to decide who best fills that role,” he said.

And his counterpart in the Senate, Sen. Richard Burr, R-North Carolina, who is leading an investigation into Russian involvement into U.S. elections, praised Flynn.

“Washington, D.C. can be a rough town for honorable people, and Flynn — who has always been a soldier, not a politician — deserves America's gratitude and respect for dedicating so much of his life to strengthening our national security. I thank him for his many years of distinguished service,” Burr said.