Graham said the resolution is co-sponsored by 41 Republicans, including Majority Leader Mitch McConnell.
"What's going on is a run around the impeachment process," Graham said at a press conference announcing the resolution, showing a poster of rights he says were afforded to Presidents Bill Clinton and Richard Nixon during their impeachment processes. "I think 41 Republican senators -- and growing -- suggests you're off script here."
Graham has frequently criticized the impeachment process in the House, while refusing to weigh in on the merits of the substance of the inquiry, which revolves around complaints from a whistleblower and other government officials who say Trump's administration pressured the Ukrainian government to investigate his political rival in return for critical security assistance funding.
Despite efforts from Graham and many other Senate Republicans to slam the process, polls show slightly more than half of Americans support the impeachment. Graham cited a July House vote on impeachment that failed 332-95, without acknowledging the months-old vote does not reflect the sea change of opinion among moderate Democrats, who now support the inquiry.
"I believe the American people were not with us on substance when it came to Clinton, but I do believe what we did very much mirrored the Watergate way of doing business," Graham said.
Earlier this week, Graham said he believed the White House needed to do a better job of messaging when it comes to impeachment. At a lunch at the White House on Thursday, Graham said acting Chief of Staff Mick Mulvaney told him he is assembling an impeachment messaging team, similar to one established by Clinton.
"He had a team that was organized, had legal minds that could understand what was being said versus the legal proceedings in question, and they were on message every day," Graham said of Clinton's impeachment team.
Graham said Clinton continued to govern during the impeachment process, which was the best thing he could have done in that situation, seemingly sending his confidante Trump some advice.
But Graham later walked back those comments, tweeting a clarification, "I did not mean to leave some with the impression the White House needed to hire a new team to handle impeachment. My interactions with the White House were in regards to a more coordinated strategy dealing with impeachment. What is in development at the White House is the emphasis on a strategy -- not personnel."
It's unclear when Graham's resolution might come up on the Senate floor for a vote or whether more moderate Republicans will also support it.