WASHINGTON—Chances of the Senate impeachment trial hearing new testimony appeared to mount on Monday in the wake of new allegations about President Trump’s motivation for freezing aid to Ukraine, as the president’s defense team argued that he was justified in pushing Kyiv to investigate Democrat Joe Biden and his son.

Former national security adviser John Bolton wrote in a draft of his forthcoming book that Mr. Trump told him in August that he wanted to freeze aid to Ukraine until the country aided investigations into Democrats,...