"I think the fundamental question for us is what is our involvement in it, and who ought to look at it," the Kentucky Republican said. "And the intelligence committee is already looking at Russian involvement in our election. As Sen. (Roy) Blunt has already indicated, it is highly likely they will want to take look at this episode as well. They have the broad jurisdiction to do it."
The Senate's second-ranking Republican and other GOP senators have called for an investigation into the episode, building on a string of investigations underway on Russian interference in the US elections. Sen. John Cornyn told reporters Tuesday that the Senate standing committees with oversight of intelligence needs to investigate.
Asked by CNN if he wanted the Senate's committees to investigate Flynn, Cornyn replied: "Yes."
But Cornyn, the Senate majority whip, was not ready to say Flynn should testify before Congress.
"I think it's symbolic of somebody with a distinguished military career making a bad mistake," Cornyn said of Flynn.
Flynn resigned from his position as national security adviser Monday following reports that the Justice Department warned the Trump administration last month that he misled administration officials about the nature of his conversations with the Russian ambassador and whether they addressed the issue of sanctions.
The conversations occurred before he was an official government employee, causing some to fear that he was potentially vulnerable to blackmail by the Russians for covering up details of the exchanges. It is illegal for unauthorized private citizens to negotiate with foreign governments on behalf of the US.
Vice President Mike Pence -- who was pushed into the center of the story after he defended Flynn on TV one month ago -- did not respond to shouted questions as he left the Capitol Tuesday. Pence has yet to say how he feels about Flynn's actions.
Pence met with Senate Republicans Tuesday at the Capitol for their weekly lunch and discussed Flynn's resignation, although senators in attendance said he did not discuss his personal feelings.
Sen. Bill Cassidy, a Louisiana Republican who used to serve with Pence in the House of Representatives, said that Pence thanked Flynn for his service and explained that "when discrepancies arose, they realized they had to do something."
"And obviously the president made the call," Cassidy added. "Folks understand that Gen. Flynn served at the discretion of the president, and the president decided to make a change."
Pence told senators they are looking at three names to replace Flynn, Sen. Bob Corker told reporters. And Sen. John McCain suggested the White House consider David Petraeus or Stanley McChrystal for the position.
Lawmakers from both the House and Senate spent much of the day Tuesday answering questions of how they would respond to Flynn's resignation and reports of his calls with the Russian ambassador.
Rep. Peter King, a New York Republican, said that Trump should be asked if he directed Flynn to talk with Russian officials, "I think that should be asked to the president. To the question of executive privilege as to the president should discuss you know what he knows."
Sen. Roy Blunt, a member of the Senate intelligence committee, told KTRS radio he wanted to speak with Flynn.
Blunt told KTRS radio
that an investigation "needs" to happen.
"I think everybody needs that investigation to happen," he said. "And the Senate intelligence committee, again that I serve on, has been given the principle responsibility to look into this, and I think that we should look into it exhaustively so that at the end of this process, nobody wonders whether there was a stone left unturned, and shouldn't reach conclusions before you have the information that you need to have to make those conclusions."
Blunt said lawmakers need to talk to Flynn "very soon."
"I would think that we should talk to Gen. Flynn very soon and that should answer a lot of questions. What did he know? What did he do? And is there any reason to believe that anybody knew that and didn't take the kind of action they should have taken," he asked.
Cornyn said that he had not been briefed on the transcripts of calls between Flynn and the Russian ambassador to the US, but expects the issue to come up with members of the Senate intelligence committee meet later Tuesday.
"I think there's an orderly process to make sure we get accurate information and we can the appropriate oversight and that's what we ought to do," the Texas senator said.
Sen. Lindsey Graham also said he wants an investigation into Flynn's conversations with a Russian ambassador about sanctions.
"I think Congress needs to be informed of what actually Gen. Flynn said to the Russian ambassador about lifting sanctions," the South Carolina Republican told CNN's Kate Bolduan on "At This Hour."
"And I want to know, did Gen. Flynn do this by himself or was he directed by somebody to do it?"
Sen. Richard Burr, chairman of the Senate Select Committee on Intelligence, told reporters he hadn't made a decision yet on whether Flynn should testify. Other Republicans have called for an investigation into the leaks over the information related to Flynn, instead of into the actions of the former national security adviser himself.
The top Democrat on the committee, Sen. Mark Warner of Virgnia, told CNN the American public "deserves" to know more about Michael Flynn's resignation and the events leading up to it.
"I think it would be appropriate for Mr. Flynn, now that he has resigned, to testify," Warner said. "I think we need to know what he knows. Not just in terms of these conversations. Clearly he indicated by his own admission that he did not fully disclose information to the vice president. I think there is more to come on this subject."
Flynn had conversations with the Russian ambassador to the United States before he was a government employee causing some to fear that he was potentially vulnerable to blackmail by the Russians.