Sen. Lindsey Graham, R-S.C., criticized a report by the New York Times that claimed President Trump "pressed" the prime minister of Australia for information to discredit the probe conducted by former Special Counsel Robert Mueller, calling it an "effort" to shut down Attorney General William Barr's investigation.

"This New York Times article about Barr talking to Australia is the beginning of an effort to shut down Barr's investigation to find out how this whole thing started," Graham said on "Hannity" Monday.

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A Justice Department official told Fox News on Monday that Barr asked Trump to make introductions to foreign countries that might have had information pertinent to U.S. Attorney John Durham's ongoing probe into possible misconduct by the intelligence community at the outset of the Russia investigation.

But, a person familiar with the situation told Fox News it would be wrong to say Trump "pressed" the Australian prime minister for information that could have discredited Mueller's now-completed probe, as the New York Times reported earlier Monday.

Graham said that Barr should be talking with Australia as well as the U.K. and Italy in order to do his job properly. He also said he would write a letter to those three countries asking them to cooperate with Barr and cited a letter his colleagues sent to Ukraine last year asking them to cooperate with Mueller or the U.S. would stop sending aid.

"So here's what I want American people to know: it's OK to cooperate with Mueller to get Trump but it's not OK to cooperate with Barr to find out if Trump was the victim of an out-of-control intelligence operation," Graham said. "We're not going to have a country like that."

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The senator told host Sean Hannity it "bothered him" that "the left" would say it was wrong for Barr to talk with other countries.

"This New York Times article is an effort to stop Barr from looking at how this whole thing began in 2016 regarding the Trump campaign," Graham said. "What are they afraid of."

Fox News' Gregg Re and Jake Gibson contributed to this report.