(CNN) Senators in both parties were considering options Tuesday to push back against President Donald Trump's decision to align himself with Russian President Vladimir Putin at their summit in Helsinki, Finland.

As Democrats sense a moment of political vulnerability for the President, Republicans are under pressure to stand by US intelligence findings that Russia meddled in the 2016 election. Hearings and additional sanctions against Russia are being floated amid bipartisan public statements of support for the US intelligence community.

Tuesday afternoon, Trump issued a statement of support for the intelligence agencies and said he accepted their findings on Russian election meddling, though he said other actors could be responsible and kept up his defense of the Helsinki meeting Wednesday morning.

Democratic leaders had a "far-ranging" discussion at a morning meeting about how the Senate should respond, according to Illinois Sen. Dick Durbin, the second-ranking Democrat in the chamber, following Trump's failure to support the US intelligence community's findings about election interference by the Russians while standing at a news conference with Putin

Democrats hope to settle on a specific proposal that they will support before the end of the day, said Durbin, who described the Democratic meeting in the Capitol.

"There are many options," Durbin told CNN. "Many of us feel we need to go beyond a statement of the obvious -- that we stand behind the professionals at the Department of Justice and intelligence community and the Department of Defense. We think the Senate has to act immediately to do something. At a minimum to have a public hearing to get to the bottom of what happened in Helsinki."

Moments later, Senate Minority Leader Chuck Schumer, a New York Democrat, speaking on the floor, called on the Senate Republican leaders to compel Trump's national security team to appear before the Senate right away to explain exactly what was said at the summit in Helsinki, especially during the one-on-one meeting between Trump and Russian President Vladimir Putin.

Democrats call for hearings, briefings

Durbin wasn't the only one to call for hearings. Democratic Sen. Jeanne Shaheen of New Hampshire called for the Senate Foreign Relations Committee to speak with the interpreter present for the Trump-Putin private meeting

"I believe the Senate Foreign Relations Committee should hold a hearing with the American interpreter who was present during President Trump and President Putin's private meeting to determine what was specifically discussed and agreed to on the United States' behalf," she said in a statement.

Democratic Sen, Mark Warner, the top Democrat on the Senate Intelligence Committee, told reporters Tuesday that senators need to do more than give statements.

"One area that I hope will be broad base consensus is we need to get a briefing as soon as possible about what happened not only in the group setting, but what I'm deeply concerned about, and warned the administration beforehand, was what happened in this one-on-one setting?" Warner said. "If the President was willing to dismiss American concerns in public, what in the heck did he say in private?"

But the second-ranking GOP leader in the Senate appeared to reject that idea when he was asked if Republicans would call on Trump officials to brief the Senate.

"I'm sure we'll find out in due course," said Sen. John Cornyn, a Texas Republican, urging patience.

Democrats are eager to push the toughest language possible in a potential "Sense of the Senate" resolution condemning Trump's comments but realize they may lose Republican support if they go too far.

"The tougher the better," said a Democrat involved in the discussions.

Durbin said he would support a resolution that conveyed the highly critical statement made by GOP Sen. John McCain, of Arizona, Monday but acknowledged Republicans might not be willing to go that far and said one Republican senator had already expressed misgivings.

Sanctions bill under consideration

Durbin said backing a tough sanctions bill authored by Florida Republican Sen. Marco Rubio and Maryland Democratic Sen. Chris Van Hollen is also under consideration. That legislation would impose sanctions on any country that interferes in US elections.

"It will take shape in the next couple of days," Rubio said about his bill. "I certainly expect our legislation to be in the mix."

"Chuck said he wanted to think things over about what was said this morning before coming up with a specific plan," said Durbin who added that Schumer is reaching out to the entire Democratic caucus on how to proceed.

Senate Majority Mitch McConnell cited Rubio's legislation as one of the options Congress is considering as a response.

"There's a possibility that we may well take up legislation related to this," McConnell said on Capitol Hill after mentioning Rubio's proposal. "In the meantime, I think the Russians need to know that there are a lot of us who fully understand what happened in 2016, and it really better not happen again in 2018."

Cornyn said he supports more sanctions.

"I think what we ought to focus on is additional sanctions instead of just some messaging exercise," said Cornyn, who pointed to a bill being written by Colorado Republican Sen. Cory Gardner as a starting point. "That was one of the things Sen. Schumer mentioned where I think we could find some common ground to turn the screws on Russia."

Gardner's legislation, introduced a few months ago, would require the State Department to determine if Russia should be designated a state sponsor of terror. Gardner explained that because of Russia's repeated bad acts around the world -- including the downing of an airline, invading of its neighbors, poisoning of individuals on foreign soil and carrying out information warfare against other countries, they should earn that designation.

"I think this is legislation Congress should consider," Gardner said. "This is one more quiver that American power can bring to bear."

Resolution also being considered

Republican Sen. Jeff Flake of Arizona said he is working on a resolution to "reaffirm support for the intelligence community," following Trump's comments.

Notably, Flake's measure is not legislation; it is a Senate resolution that would not become law. Flake said his goal is to bring the proposal to the Senate floor and have it pass with unanimous consent this week or next. Flake said he has spoken with Senate Majority Leader Mitch McConnell about the resolution but stopped short of saying that the Kentucky Republican supported the move.

"They're not saying don't do this or this is the wrong thing to do," Flake said.

Durbin said he hopes Democrats and Republicans can reach a bipartisan agreement, as they did previously when they voted to impose sanctions against Russia for election interference.

"We hope they will join us in a bipartisan effort," Durbin said. "They did on the imposition of sanctions despite reluctance from the White House. So, there is some sentiment on the Republican Senate side against Vladimir Putin and maybe we'll see it again."

Tennessee Republican Sen. Bob Corker, who chairs the Foreign Relations Committee, said he thinks the Senate should act first on an anti-tariffs measure to show Putin the chamber will defend European allies. Corker also warned Tuesday that Trump's actions wouldn't easily be fixed by congressional action.

"There is no question that the President's 15 minutes at a press conference can do more damage than it takes us months to overcome. I would say the first step we should take would be to pass this tariff legislation," Corker said.

This story has been updated with additional developments.