Trump lawyer Rudy Giuliani's claim that special counsel Robert Mueller is hoping to end his investigation into whether the president obstructed justice in the Russia probe by Sept. 1 is "entirely made up," a new report says.

A U.S. official familiar with the case said Giuliani's assertion in a New York Times article on Sunday about Mueller's supposed target date was "another apparent effort to pressure the special counsel to hasten the end of his work," Reuters reported.

"He'll wrap it up when he thinks he's turned every rock," the unidentified source said, referring to Mueller's inquiry into possible obstruction by President Donald Trump into the question of Russian meddling in the 2016 presidential election.

"And when that is will depend on how cooperative witnesses, persons of interest and maybe even some targets are, if any of those emerge and on what new evidence he finds."

The source added that Mueller's investigation into possible obstruction — which is just one facet of his ongoing probe — will not end based on "some arbitrary, first-of-the-month deadline on of the president's attorneys cooks up."

The source's comments to Reuters came a day after Giuliani was quoted in the Times as saying that Mueller's office told him about two weeks ago that it hopes to finish the obstruction investigation by Sept. 1.

Peter Carr, a spokesman for Mueller's office, declined to comment when asked about the reports by CNBC.

Jay Sekulow, another lawyer for Trump, could not immediately be reached for comment by CNBC.

Since being hired by Trump in April, Giuliani has been an outspoken defender of the president. He has pushed to end to Mueller's inquiry into the president; he's said that the special counsel's office has told him it cannot criminally charge Trump; and he has insisted on a series of conditions for any interview Trump may give the special counsel.

Giuliani and Trump strongly deny that the president has obstructed the probe into Russian interference in the American election.

On Sunday, Trump unleashed an angry tweetstorm over the Mueller inquiry, referring to it yet again as a "witch hunt," and demanding that the Justice Department investigate whether the department and FBI "infiltrated and surveilled the Trump Campaign for Political Purposes."

Trump fired James Comey last year after the president suggested that the then FBI director drop an investigation into Trump's first national security advisor, Michael Flynn.

Trump has said that when he decided to fire Comey "I said to myself, I said, 'You know this Russia thing with Trump and Russia is a made-up story.' "

Giuliani recently said that Trump fired Comey because the FBI director "would not — among other things — say that he [Trump] wasn't a target of the investigation."

Mueller was appointed a year ago, May 17, 2017, as special counsel to investigate the Russian operation on the heels of Comey's firing.

Flynn since has pleaded guilty to lying to the FBI about his postelection contacts with Russia's ambassador to the United States. He is now cooperating with Mueller, who brought the charge against him.