"If they can satisfy Rudy, they can satisfy the President," said a person familiar with the meeting.
CNN contacted the White House for comment.
A key accusation in the whistleblower's complaint that prompted the impeachment probe into the President's reports in the Ukraine is that Giuliani, a private citizen, had presented a different American policy to that of US diplomats in Ukraine.
Their goal was to tell Trump that they had a favorable impression of Zelensky and his government and that he was a reformer that Trump was supposed to trust and commit to, according to three sources who were familiar with the meeting .
The sources said they hoped to organize a meeting between Trump and Zelensky. They believed that the Ukraine under Zelensky was a more trusted ally than the previous Ukrainian regimes and that a visit between Trump and Zelensky could prove to the Russian government that the United States has embraced a free Ukraine, according to two of the sources.
Perry led the delegation that had participated in the Zelensky inauguration and was a central figure in the May meeting with Trump.
It has become clear to Trump administration officials, according to sources, that they should have dealt with Giuliani.
Volker mentioned last week in speaking about how central Giuliani was in the President's approach to foreign policy in Ukraine.
"The president was very skeptical," Volker told House committees, describing what happened when he, Sondland and Perry talked to Trump. "During that conversation, he referred to conversations with Mayor Giuliani."
US officials then decided to correct the information feed that comes to the President as they realize that they should work with and around Giuliani, according to sources.
Perry acknowledged talking to Giuliani, according to a spokesman for the Department of Energy. The spokesman did not comment on the meeting itself or the substance of the conversations with Giuliani.
When asked if he had ever spoken to Perry about Ukraine, Giuliani responded by rejecting the question.
"Henry Kissinger never called me for Ukraine," Giuliani wrote on CNN Tuesday. "What about Colin Powell."
He then said that any information about his possible discussions with Perry would be privileged. He did not explain why.
CNN's Michael Warren, Rene Marsh and Jeremy Diamond contributed to this report.