Senator John Thune of South Dakota, the No. 2 Republican, said Thursday that it was “hard to say” whether or how often the Trump impeachment trial might move into closed session. But gathering privately could occur as early as Tuesday, depending on how seriously Republicans and Democrats clash over ground rules for the remainder of the proceeding.

Senators from both parties said they would like the trial to be as open as possible — and are well aware of the troubling optics of shuttering the Senate while weighing the future of the president.

“I hope none of it is closed,” said Senator John Cornyn, Republican of Texas.

That view might be shared by Mr. Trump, who has suggested that he is counting on a public trial in the Republican-controlled Senate to discredit the impeachment initiated by House Democrats and vindicate him, proving his frequent claim that he did nothing wrong in his dealings with Ukraine.

But given deep disputes between Republicans and Democrats over what the trial should entail and a prohibition on senators talking on the floor, Senate officials said that some closed sessions were probably inevitable. They are quietly laying the groundwork for that option, hoping to tamp down any criticism that information is being hidden.