WASHINGTON—Republican leaders said they don’t currently have enough votes to block witnesses in President Trump’s Senate impeachment trial, people familiar with the matter said, after his legal team concluded its efforts to counter Democrats’ charges that the president abused power and obstructed Congress.

On the third and final day of presentations by the Trump legal team, lawyers tried to cast doubts on the importance and credibility of allegations by former national security adviser John Bolton about the president’s motives for freezing aid to Ukraine. Republicans had hoped to wrap up the trial with an acquittal of the president by this week, but Democrats have said Mr. Bolton should appear under oath to offer a firsthand account of the president’s motivations for freezing aid to Ukraine—a matter at the heart of the impeachment case.

At a meeting of all Republican senators late Tuesday, Senate Majority Leader Mitch McConnell (R., Ky.) said the vote total wasn’t where it needed to be on blocking witnesses or documents, the people familiar with the matter said. He had a card with “yes,” “no,” and “maybes” marked on it, apparently a whip count, but he didn’t show it to senators.

Sens. Cory Gardner of Colorado, Martha McSally of Arizona and Thom Tillis of North Carolina, who face competitive races in the fall, addressed their colleagues in the meeting, people familiar with the matter said. Mr. Gardner said a longer trial would lead to more Democratic attacks, according to a spokesman, and Mr. Tillis called impeachment a sham. Ms. McSally’s office said she doesn’t comment on what happens in private meetings.

An administration official said the White House was optimistic it would get the necessary votes by Friday. “We are still in the game,” the official said.