WASHINGTON — President Trump pressed the Ukrainian president in a July call to investigate former Vice President Joseph R. Biden Jr.’s son, according to a person familiar with the conversation, an apparently blatant mixture of foreign policy with his 2020 re-election campaign.

Mr. Trump also repeatedly told the Ukrainian leader, Volodymyr Zelensky, to talk with his personal lawyer Rudolph W. Giuliani, who had been urging the government in Kiev to investigate Mr. Biden and his family, according to two other people briefed on the call.

Mr. Trump’s request is part of the secret whistle-blower complaint that is said to be about Mr. Trump and at least in part about his dealings with Ukraine, according to two people familiar with the complaint.

Mr. Biden is a leading candidate for the Democratic presidential nomination. The president has made no secret that he wanted Ukraine to investigate any improper overlap between Mr. Biden’s own diplomatic efforts there and his son’s role with a gas company owned by a Ukrainian oligarch. “Someone ought to look into Joe Biden,” Mr. Trump told reporters on Friday in response to a question about his call with Mr. Zelensky.