"Congress must act, right away, to check this potential Presidential abuse of power," said Sen. Richard Blumenthal (D-CT), who is backing Graham's bill.
A major undercurrent of these two bills - and of a bill that strengthened sanctions against Russia - is a desire by both parties in Congress to exert more Constitutional authority when it comes to lawmakers being a check on the power of the Executive.
That tweet by the President early on Thursday morning - which blamed Congress for bad relations with Russia and Vladimir Putin - drew a derisive reaction from some GOP lawmakers, who have become more and more vocal in recent days about their differences with the President on the Moscow regime.
"Huh," said a puzzled looking Sen. Jeff Flake (R-AZ). "It is what it is," he added, with a head-shaking chuckle.
"The relationship we have with Russia is solely because of Putin," said Sen. Bob Corker (R-TN), who has repeatedly said he is proud of the Congress for its work on the Russia sanctions bill, which Mr. Trump clearly did not like, but signed into law anyway.
But not everyone was at odds with the President when it comes to the Russia investigation.
"Robert Mueller is in a position where he has undermined his ability to carry out the law and must resign," said Rep. Trent Franks (R-AZ), though that view is clearly in the minority for GOP lawmakers.