A Trump ally, the Senate Judiciary Committee chairman Graham tweeted on Thursday that the resolution has more than 40 cosponsors, including Senate Majority leader Mitch McConnell, reports Xinhua news agency.
Graham called the current proceedings "illegitimate and dangerous", while urging the House to vote to initiate a formal impeachment inquiry, provide Trump with what he called "due process", and allow Republicans to fully participate.
House Speaker Nancy Pelosi announced the initiation of a formal impeachment inquiry into the President on September 24 over his conduct during a controversial phone call with his Ukrainian counterpart Volodymyr Zelensky on July 25.
The impeachment inquiry was prompted by a whistleblower complaint filed in late August alleging that Trump pressured Zelensky to investigate Democratic presidential candidate Joe Biden and his son Hunter, whom the US President has accused of involvement in a corruption case in Ukraine.
The White House had said that they would not provide documents or witnesses to impeachment investigators because it considers the investigation to be unfair and illegitimate.
Trump has denied any wrongdoing and had urged more efforts from Republicans to defend him and challenge the impeachment inquiry.
The resolution came a day after dozens of Republicans stormed a closed-door hearing for a Pentagon official as part of the impeachment inquiry.
( With inputs from IANS )