WASHINGTON—The Trump administration sought to use a potential meeting between the president and his Ukrainian counterpart as leverage to press Kyiv to investigate Joe Biden, newly released text messages indicated, as President Trump called on China to also investigate his political rival.

The president’s efforts to persuade Ukraine in a July phone call to investigate Mr. Biden have already set off an impeachment inquiry by House Democrats, who are looking at whether the president abused the power of his office for political...