WASHINGTON—Senators wavering on calling new witnesses in President Trump’s impeachment trial hinted Wednesday at the factors they were considering, as Republican leaders tried to persuade lawmakers to bring the proceedings to a close later this week without fresh testimony.

After several days of opening arguments by each side that offered little give-and-take, Wednesday’s session brought an opportunity for senators to direct questions at Mr. Trump’s legal team and Democratic House members, who are serving as prosecutors in...