A draft of Graham's resolution obtained by NBC News says the Constitution does not provide Pelosi, D-Calif., "with the power to effectively veto a resolution passed by a duly elected majority of the House of Representatives by refusing to transmit such a resolution to the Senate." Such a withholding of the articles "is a flagrant violation of the separation of powers expressly outlined in the bicameral impeachment process under the Constitution of the United States," it says.
On Sunday, Graham, who chairs the Judiciary Committee, proposed going further, telling Fox News, "If we don't get the articles this week, then we need to take matters in our own hands and change the rules."
That would allow the Senate to begin the trial without the articles being submitted, he told the network's "Sunday Morning Futures."
"My goal is to start this trial in the next coming days, not let [House Speaker] Nancy Pelosi take over the Senate," Graham said.
Graham's current resolution does not go so far as to call for a change in Senate impeachment rules, although Kevin Bishop, his communications director, told NBC News that it does not "preclude" what the senator discussed Sunday.
It was not immediately clear when Graham would introduce the resolution.
Pelosi is holding firm in her stance not to submit the articles until the Senate process is explicitly laid out, though she suggested she could send them in the near future.
"Soon, the Senate will have the opportunity to honor its oath to 'do impartial justice according to the Constitution and laws,'" Pelosi wrote in a letter to colleagues on Tuesday before arguing the process proposed by Senate Majority Leader McConnell, R-Ky., is unfair.
No trial can begin until the articles are sent.
McConnell has said he wants the Senate to adhere to the process used during former President Bill Clinton's impeachment trial in 1999, which included an initial agreement to hear the case followed by a later vote to hear witness testimony. On Tuesday, McConnell, who last month pledged "total coordination" with the White House on the trial, said he has the votesto proceed with his plan.
Meanwhile, Senate Minority Leader Chuck Schumer, D-N.Y., has emphasized that in Clinton's impeachment trial, the witnesses had already given grand jury testimony that was available to senators, but in Trump's case, several key administration witnesses have refused to testify.
"The Clinton model is totally different," Schumer told ABC News' George Stephanopoulos on Sunday when asked about his disagreement with McConnell, adding, "One, they had been heard from already. Every one of those witnesses had been heard from before."
The minority leader said he wants the Senate to call four witnesses to testify about Trump's conduct toward Ukraine, including former national security adviser John Bolton, who announced this week he would testify if subpoenaed, and acting White House chief of staff Mick Mulvaney.
Earlier this week, Sen. Josh Hawley, R-Mo., introduced a resolution to change Senate impeachment rules so that the chamber could vote on dismissal of the charges if the articles were not transmitted within 25 days after adoption. So far, 12 Republicans have cosponsored that proposal.
Other Republicans have cast doubt on the idea that they would change Senate rules, with Sen. Roy Blunt, of Missouri, a member of Senate GOP leadership, telling The Washington Examiner the likelihood of such changes is "remote."