One is the origin of the special counsel’s investigation into Russia’s interference in the 2016 election. The other is the involvement of former Vice President Joseph R. Biden Jr.’s son in a gas company owned by a Ukrainian oligarch.

Mr. Giuliani’s plans create the remarkable scene of a lawyer for the president of the United States pressing a foreign government to pursue investigations that Mr. Trump’s allies hope could help him in his re-election campaign. And it comes after Mr. Trump spent more than half of his term facing questions about whether his 2016 campaign conspired with a foreign power.

“We’re not meddling in an election, we’re meddling in an investigation, which we have a right to do,” Mr. Giuliani said in an interview on Thursday when asked about the parallel to the special counsel’s inquiry.

“There’s nothing illegal about it,” he said. “Somebody could say it’s improper. And this isn’t foreign policy — I’m asking them to do an investigation that they’re doing already and that other people are telling them to stop. And I’m going to give them reasons why they shouldn’t stop it because that information will be very, very helpful to my client, and may turn out to be helpful to my government.”