Donald Trump wants William Barr to consider investigating Joe Biden, a potential opponent in next year’s presidential election, over Biden’s ties to Ukraine, Trump’s personal lawyer said on Sunday.

The former New York mayor Rudy Giuliani made the remark as fears grew among Democrats that the president will use the US justice department – and even help from foreign officials – to go after his political rivals.

Giuliani told the Guardian he and Trump agreed that the attorney general should decide whether US investigators ought to look into a lucrative business deal in Ukraine obtained by Biden’s son while Biden was vice-president.

“I believe the president has the same view that I have – that the attorney general should make this call,” Giuliani said.

Hailing Barr as “independent, brilliant and honest”, Giuliani added: “I can’t tell Attorney General Barr what to do. He’s a very, very fine lawyer. My experience with him is that he’s a very honorable man, and he will do the right thing.”

Several Democratic candidates for the party’s presidential nomination have raised alarms about Giuliani’s efforts to attack Biden. Giuliani had planned to travel to Ukraine to seek information from senior officials in Kiev, but cancelled the trip amid sharp criticism.

Adam Schiff, chair of the House intelligence committee, told ABC’s This Week earlier on Sunday: “Going after [Biden’s] son is just a method of going after someone the president believes is his most formidable opponent.”

Giuliani has been working to resurface allegations that Biden improperly used his position as Barack Obama’s “point man” on Ukraine to help the business career of his younger son, Hunter, a corporate lawyer and former reservist in the US navy.

Biden has publicly boasted that in 2015 he threatened to withhold US aid to Ukraine, as a way to help oust the country’s top prosecutor. Giuliani claims Biden did so because the prosecutor had been investigating a gas company that was paying Hunter Biden as a director. The investigation was dropped.

The Bidens strenuously deny any connection. They are supported by prominent anti-corruption campaigners in Ukraine, who say the prosecutor, Viktor Shokin, was simply failing to thoroughly investigate corruption among the country’s elite and deserved to be fired.

In 2015, when questions about the situation first arose, a spokeswoman for the Bidens said: “Hunter Biden is a private citizen and a lawyer. The vice-president does not endorse any particular company and has no involvement with this company.”

No solid evidence has been produced by Giuliani or his allies to suggest Biden took action to benefit his son.

A former senior official in Shokin’s office told Bloomberg News last week the Bidens did not interfere with the investigation into the gas company, Burisma Holdings. In fact, inquiries were dormant long before Biden made his ultimatum for Shokin to be fired, the former official said.

Records indicate that even after Hunter Biden’s appointment, US and British officials were pressuring the prosecutor’s office to more vigorously investigate Burisma, which was controlled by a minister in Ukraine’s former pro-Russian administration.

Donald Trump speaks at the White House on Thursday. Photograph: UPI/Barcroft Images

Trump said in an interview with Politico on Friday that the allegation against Biden “could be a very big situation” and insisted that “certainly it would be an appropriate thing” for him to discuss a potential investigation with Barr, but that he had not yet done so.

On Sunday, Giuliani tried to dismiss allegations of partisanship, noting there remained 17 months before voters go to the polls and arguing that the issue should be examined “sooner rather than later” to avoid clashes with the presidential campaign.

“My experience says you must investigate it now so you don’t run into it four months before the election like they did with Hillary,” he said, recalling the FBI’s investigation into Hillary Clinton’s handling of classified information over a private email server, which rocked her 2016 campaign against Trump.

Trump and his team led chants of “lock her up” aimed at Clinton during campaign rallies in 2016, drawing condemnation.

The report published last month by Robert Mueller, the special counsel, said Trump asked his first attorney general, Jeff Sessions, “to take a look at” investigating Clinton after they entered office.

Barr’s handling of the Mueller report has infuriated Democrats in Congress, who have accused him of protecting Trump.