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...