WASHINGTON — Even as Democrats intensified their scrutiny this week of Rudolph W. Giuliani’s role in the pressure campaign against the Ukrainian government that is at the heart of the impeachment inquiry, Mr. Giuliani has been in Europe continuing his efforts to shift the focus to purported wrongdoing by President Trump’s political rivals.

Mr. Giuliani, the president’s personal lawyer, met in Budapest on Tuesday with a former Ukrainian prosecutor, Yuriy Lutsenko, who has become a key figure in the impeachment inquiry. He then traveled to Kyiv on Wednesday seeking to meet with other former Ukrainian prosecutors whose claims have been embraced by Republicans, including Viktor Shokin and Kostiantyn H. Kulyk, according to people familiar with the effort.

The former prosecutors, who have faced allegations of corruption, all played some role in promoting claims about former Vice President Joseph R. Biden Jr., a former United States ambassador to Ukraine and Ukrainians who disseminated damaging information about Mr. Trump’s campaign chairman, Paul Manafort, in 2016.

Those claims — some baseless and others with key disputed elements — have been the foundations of the effort by Mr. Trump and Mr. Giuliani to pressure the Ukrainian government to commit itself to investigations that would benefit Mr. Trump heading into his re-election campaign. That effort in turn has led to the impeachment proceedings in the House against the president.