Conservative Senator Mike Lee (R-Utah) said President Trump should nominate Judge Merrick Garland to replace James Comey as Federal Bureau of Investigation director.
The White House had no immediate comment.
That position puts him opposite of President Donald Trump's administration, which laid out a proposal outside groups have said adds to the deficit, despite administration officials arguing that tax reform would pay for itself with economic growth.
"The president has a chance to clean up the mess he mostly created", Graham said, adding: "I have no evidence that the president colluded with the Russians at all. but we don't know all the evidence yet".
Senate Majority Leader Mitch McConnell politely asked Tuesday that the endless stream of problems and controversy from the White House needs to stop. If Trump refuses, Schumer says Americans will doubt that their president is capable of safeguarding critical secrets. "The Senate Intelligence Committee should be briefed on this important issue immediately".
Addressing the controversy over last week's firing of FBI Director James Comey, the Kentucky Republican said he had recommended as a potential replacement President Barack Obama's former Supreme Court nominee Merrick Garland.
Republican Senators Mike Lee and Orrin Hatch, both from Utah, were among the first to float Garland's name as someone who could conceivably take Comey's job. One White House aide called for the problematic portion of Trump's discussion to be stricken from internal memos and for the full transcript to be limited to a small circle of recipients, efforts to prevent sensitive details from being disseminated further or leaked.
Gee, why would McConnell think this was such a good idea? Another Republican whose name had been mentioned as a possible candidate, Rep. Trey Gowdy of SC, said Monday that he also had taken himself out of the running. So they are trying to do more of that. John McCain, who called the reports "deeply disturbing".
Sen. Bob Corker of Tennessee, who was a contender to be Trump's secretary of state, was one of the first Republicans to react to the news, saying the reports were "worrisome".
Lee backed the idea again on "Fox News Sunday".
After the reports spread Monday night, Speaker Paul Ryan, R-Wisconsin, released a statement through his spokesman, Doug Andres.
Rep. Mike Gallager, a Wisconsin Republican in his first term, tweeted "For the goal of transparency, the White House should share a transcript of the meeting with the House and Senate intelligence committees".
The developing story is expected to consume Congress's attention, particularly as House members return later Tuesday from a week-long recess.