WASHINGTON — Months after backing out of a trip to Ukraine amid criticism that he was mixing partisan politics with foreign policy, Rudolph W. Giuliani, President Trump’s personal lawyer, has renewed his push for the Ukrainian government to pursue investigations into political opponents of Mr. Trump.

Over the last few weeks, Mr. Giuliani has spoken on the phone and held an in-person meeting, in Madrid, with a top representative of the new Ukrainian president, encouraging his government to ramp up investigations into two matters of intense interest to Mr. Trump.

One is whether Ukrainian officials took steps during the 2016 election to damage Mr. Trump’s campaign. The other is whether there was anything improper about the overlap between former Vice President Joseph R. Biden Jr.’s diplomatic efforts in Ukraine and his son’s role with a gas company there.

Mr. Giuliani said he was acting on his own as a private citizen, with the knowledge and assistance of the State Department. He would not say whether Mr. Trump approved — or is aware of — the effort.