President Trump in a July phone call repeatedly pressured the president of Ukraine to investigate Joe Biden’s son, according to people familiar with the matter, urging Volodymyr Zelensky about eight times to work with Rudy Giuliani on a probe that could hamper Mr. Trump’s potential 2020 opponent.

“He told him that he should work with [Mr. Giuliani] on Biden, and that people in Washington wanted to know” if his lawyer’s assertions that Mr. Biden acted improperly as vice president were true, one of the people said. Mr. Giuliani has suggested Mr. Biden’s pressure on Ukraine to fight corruption had to do with an investigation of a gas company for which his son was a director. A Ukrainian official this year said he had no evidence of wrongdoing by Mr. Biden or his son Hunter Biden.

Mr. Trump in the call didn’t mention a provision of U.S. aid to Ukraine, said this person, who didn’t believe Mr. Trump offered the Ukrainian president any quid pro quo for his cooperation on any investigation.

The interactions between the president, Mr. Giuliani and Ukraine have come under scrutiny in recent days in the wake of a whistleblower complaint that a person familiar with the matter said involves the president’s communications with a foreign leader. The complaint, which the Washington Post reported centers on Ukraine, has prompted a new standoff between Congress and the executive branch.

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Separately, lawmakers are investigating any connection between the review of foreign aid to Ukraine and the efforts to pressure Kiev to look into Mr. Biden.

Mr. Giuliani in June and August met with top Ukrainian officials about the prospect of an investigation, he said in an interview. After the July call between the two presidents, the Ukrainian government said Mr. Trump had congratulated Mr. Zelensky on his recent election and expressed hope that his government would push ahead with investigations and corruption probes that had stymied relations between the two countries.

The White House declined to comment.

Mr. Biden, in a statement Friday, called for the White House to release the transcript of the president’s call with Mr. Zelensky.

“Such clear-cut corruption damages and diminishes our institutions of government by making them tools of a personal political vendetta,” he said.

Mr. Trump only recently emerged from the nearly two-year investigation by Robert Mueller into whether his campaign sought help in the 2016 election from a different country: Russia. While Mr. Mueller said in his report this spring that he didn’t establish a conspiracy between Moscow and the Trump campaign, Mr. Trump’s efforts to seek Ukraine’s help in damaging a potential political opponent are certain to revive criticism that the president welcomes campaign help from foreign countries.

Mr. Trump on Friday defended his July call with Mr. Zelensky as “totally appropriate” but declined to say whether he had asked the Ukrainian leader to investigate Mr. Biden. At the same time, he reiterated his call for an investigation into Mr. Biden’s effort as vice president to oust Ukraine’s prosecutor general. “Somebody ought to look into that,” he told reporters.

In a series of tweets on Saturday morning, he again defended his call while taking aim at Mr. Biden. “Nothing was said that was in any way wrong,” Mr. Trump said of his conversation with Mr. Zelensky. “But Biden’s demand, on the other hand, was a complete and total disaster.”

In recent months, Mr. Giuliani has mounted an extensive effort to pressure Ukraine to do so. He said he met with an official from the Ukrainian prosecutor general’s office in June in Paris, and met with Andriy Yermak, a top aide to Mr. Zelensky, in Madrid in August. Mr. Giuliani said in an interview this month that Mr. Yermak assured him the Ukrainian government would “get to the bottom” of the Biden matter.

The August meeting came weeks before the Trump administration began reviewing the status of $250 million in foreign aid to Ukraine, which the administration released earlier this month. Mr. Giuliani said he wasn’t aware of the issue with the funds to Ukraine at the time of the meeting.

He said his meeting with Mr. Yermak was set up by the State Department, and said he briefed the department on their conversationlater. The State Department had no immediate comment.

In late August, after Mr. Trump canceled a planned trip to Poland where he had been scheduled to meet with Mr. Zelensky, Mr. Giuliani said Mr. Yermak called him to ask whether the cancellation had anything to do with Mr. Zelensky, which Mr. Giuliani assured him it did not. Mr. Trump at the time said he needed to deal with Hurricane Dorian.

Mr. Trump is to meet with Mr. Zelensky in person for the first time next week, at the United Nations General Assembly gathering in New York.

Michael Atkinson, the Trump-appointed inspector general of the intelligence community, met Thursday morning with the House Intelligence Committee in a closed session to discuss the whistleblower complaint. Mr. Atkinson declined to tell lawmakers the substance of the complaint or whether it involves the president, but he did say it involves more than one episode and is based on a series of events, according to several people who attended or were briefed on the meeting.

Joseph Maguire, a retired Navy vice admiral serving as the acting director of national intelligence, is to appear before both the Senate and House intelligence committees next week about the complaint, though it remains unclear if he will be willing to divulge details about its underlying substance.

Stymied Democrats in Congress continued to mull potential avenues to obtain the complaint. Rep. Adam Schiff (D., Calif.), chairman of the House Intelligence Committee, said he was considering a lawsuit to obtain the complaint or withholding funding from the Office of the Director of National Intelligence. Mr. Schiff has accused Mr. Maguire of violating the law by not sending the complaint to Congress, as required under the federal whistleblower statute.

“It’s been very hard for the director of national intelligence to explain why he is the first ever in that position to withhold an urgent whistleblower complaint from Congress,” Mr. Schiff said Friday.

Mr. Maguire’s office consulted the Justice Department’s Office of Legal Counsel, which determined that the allegation didn’t meet the statutory definition of an “urgent concern” requiring reporting to the intelligence committees, the Justice Department said.

Typically, opinions from the OLC are seen as binding on the executive branch, legal experts said, and it remains unclear how or whether Mr. Maguire could transmit the complaint to lawmakers now.

Even before the debate over the whistleblower complaint, Democratic lawmakers had begun investigating interactions with Ukraine by the president and his lawyer. Earlier this month, the House Foreign Affairs, Intelligence and Oversight committees sent letters to the White House and State Department seeking records of interactions involving the president and Mr. Giuliani and the Ukrainian government.

In the interview this month, Mr. Giuliani said he had sought in the spring to meet with Mr. Zelensky—at the time Ukraine’s president-elect—and planned a trip to Kiev to pressure the Ukrainian government to pursue two investigations: one into whether Ukraine, under its previous leader, had sought in 2016 to hurt the Trump campaign and bolster his opponent; and another into diplomatic efforts in the country by Mr. Biden, who is currently leading the Democratic presidential field.

Mr. Giuliani ultimately canceled that trip after his plan was made public. Mr. Trump was aware of the planned meeting, he said.

Mr. Biden as vice president made several trips to Ukraine to press the government to root out widespread corruption. That included seeking the ouster of former prosecutor general Viktor Shokin, who had investigated a private Ukrainian gas company, Burisma Group, of which Hunter Biden was a board member. Mr. Giuliani has suggested Mr. Biden’s motivation was to protect his son, a lawyer who has been involved in several investment and consulting firms, although Mr. Shokin had already completed his investigation of Burisma Group before he left office.

Yuriy Lutsenko, Ukraine’s prosecutor general at the time, told Bloomberg News in May he had no evidence of wrongdoing by Mr. Biden or his son.

In an interview Thursday evening, Mr. Giuliani said he wasn’t aware whether the whistleblower complaint related to Ukraine. But in a Twitter post later that evening, he defended the possibility that Mr. Trump had urged Mr. Zelensky to investigate his potential campaign opponent.

“A President telling a Pres-elect of a well known corrupt country he better investigate corruption that affects US is doing his job,” Mr. Giuliani wrote.

Mr. Giuliani said earlier this month that Mr. Trump likely would raise the Biden matter with Mr. Zelensky when they meet, saying the matter was “on his mind.” A senior administration official said Friday that the two would discuss how to expand energy cooperation and trade ties.

Write to Alan Cullison at alan.cullison@wsj.com, Rebecca Ballhaus at Rebecca.Ballhaus@wsj.com and Dustin Volz at dustin.volz@wsj.com