WASHINGTON—President Trump asked his Ukrainian counterpart, Volodymyr Zelensky, to “look into” former Vice President Joe Biden and his son and said he would direct his personal lawyer and Attorney General William Barr to contact Mr. Zelensky to help him in a possible investigation, according to a document released Wednesday by the White House.

The document, a rough transcription of a July phone call between the leaders, also revealed that Mr. Trump, before asking Ukraine to examine actions by Mr. Biden’s son, reminded Mr....