“We’re at Cohen’s mercy for the dates,” Mr. Giuliani said, adding that the president “doesn’t remember the dates. He does remember conversations about Moscow. He does remember the letter of intent. He does remember, after that, fleeting conversations.”

He added that he was trying to keep Mr. Trump from legal exposure if prosecutors uncover evidence of a conversation that the president has said he does not recall.

Like so many other threads of the Russia saga, the story that the president’s aides have told about the Trump Tower Moscow negotiations has changed repeatedly. First, they said that the discussions never moved beyond their infancy, barely involved Mr. Trump, and ended well before the Republican primaries. Then, when Mr. Cohen pleaded guilty in November to lying to Congress about the Tower negotiations, the special counsel’s office revealed they extended at least until the middle of 2016.

When — and even if — the discussions officially ended remains unclear, and Mr. Giuliani did not provide clarity.

Even after Mr. Trump secured a surprise victory in November 2016 — and as evidence was mounting that the Russians had carried out a sophisticated campaign to disrupt the presidential election — Mr. Trump’s top aides took part in numerous meetings and phone conversations with Russians that have been a focus of Mr. Mueller’s investigation.

Jared Kushner, the president’s son-in-law and senior adviser, met with the head of a Russian bank under sanctions and asked Russia’s ambassador to Washington, Sergey I. Kislyak, whether Mr. Trump’s aides could use phone lines at the Russian embassy to communicate with Moscow during the presidential transition. Michael T. Flynn, who would become President Trump’s first national security adviser, discussed sanctions with Mr. Kislyak numerous times in December 2016 as President Obama punished Russia for its campaign of election interference.

Mr. Flynn pleaded guilty a year later to lying to F.B.I. agents about his communications with Mr. Kislyak.