But the weeks to come will be anything but normal, as the Senate opens only the third presidential impeachment trial in U.S. history.
McConnell's draft resolution proposes giving both the prosecution and defence teams 24 hours over two days for their respective arguments.
They called the two articles of impeachment - approved largely along party lines in the Democratic-controlled House - the product of "a rigged process" and "constitutionally deficient on their face" because they involved no violation of established law.
Senate Minority Leader Chuck Schumer accused McConnell of casting aside public statements that he would use the same rules as under the Clinton trial, adding that the majority leader is clearly "hell-bent on making it much more hard to get witnesses and documents and intent on rushing the trial through".
In recent months, Stefanik - a member of the House Intelligence committee - has emerged as one of Trump's fiercest champions, appearing frequently in conservative media. They say he then obstructed the House investigation into his actions, but Trump's lawyers contend he did "absolutely nothing wrong" and is a victim of overzealous House Democrats.
Two other presidents - Andrew Johnson in 1868 and Bill Clinton in 1998 - were impeached by the House but acquitted in Senate trials and remained in office.
McConnell said the Senate would decide whether witnesses will be permitted in the trial.
The change would represent a departure from the parameters of the Clinton impeachment process, in which the evidence obtained from the House of Representatives was automatically included in the Senate.
Mr Romney is among a small number of Republican senators who want to consider witness testimony and documents that weren't part of the House impeachment investigation.
"There's people.in the House, that I know who are not necessarily insane about President Trump, but if that'd been an open ballot, a secret ballot, it wouldn't have mattered, they still felt the charges against him did warrant impeachment", King told Kelly.
In a six-page filing formally responding to the House impeachment charges submitted shortly after and filled with partisan barbs against House Democrats, Mr. Trump's lawyers denounced the case as constitutionally and legally invalid, and driven purely by a desire to hurt Mr. Trump in the 2020 election.
House Democrats in their initial court filing over the weekend called Trump's conduct the "worst nightmare" of the framers of the Constitution.
They attempted to dismiss the second article of impeachment, that Mr Trump obstructed Congress, by arguing that he was fairly using his executive powers when rejecting legal demands for White House officials to testify to the impeachment investigation.
While the Senate, with a 53-47 Republican majority, is not at all expected to mount the two-thirds voted needed for conviction, the president's lawyers went so far as to suggest such an outcome would be an "unconstitutional conviction" because of the overly broad articles of impeachment from the House.
It is claimed that, during a call with Ukrainain President Volodymyr Zelensky, he held back military aid in exchange for an investigation into Hunter Biden - the son of Trump's political rival, Joe Biden, and a former member of the board of Ukrainian energy firm Burisma.
Democrats say Trump's behavior is exactly what the country's founding fathers had in mind when they created the mechanism to accuse and remove a president from power.
No president has ever been removed by the Senate. Neither Trump nor his allies have provided evidence to support them, and Biden has denied them.