But Pompeo declared he did not want "to talk about any of the facts", while President Donald Trump said innocence must be presumed, drawing a parallel with his US Supreme Court judge nominee who faced sexual assault accusations.
"This guy is a wrecking ball, he had this guy murdered in a consulate in Turkey, and to expect me to ignore it, I feel used and abused", Graham said on "Fox and Friends".
Nauert, also the acting under-secretary for public diplomacy and public affairs, accompanied Secretary of State Mike Pompeo on a trip to Turkey and Saudi Arabia this week to meet with the country's leaders to discuss journalist Jamal Khashoggi's disappearance.
Asked about a recording described by the Turkish newspaper Yeni Safak, Trump said, "We've asked for it, if it exists".
"I'll have a full report on that from Mike (Pompeo) when he comes back".
"You don't know whether or not we have, do you?" he asked, before adding, "I'm not going to tell you".
Trump, who has forged closer ties with Saudi Arabia and the 33-year-old crown prince, said the USA has asked Turkey for any audio or video evidence.
His comments followed publication in pro-government Turkish media of allegations purporting to confirm that Khashoggi was not just murdered by Saudi agents in their consulate in Istanbul, but dismembered.
An 11-member Saudi investigation team has arrived at the Saudi Consul's Istanbul residence ahead of an anticipated search by Turkish police in connection with journalist Jamal Khashoggi's disappearance, CNN Turk reported.
"We will probably know that by the end of the week", Trump said.
The New York Times reported that a suspect identified by Turkey was a frequent companion of the prince's.
On Tuesday, The Washington Post, for which Khashoggi had been writing, accused the Trump administration of performing a "diplomatic cleanup operation" for Riyadh.
"Well, he wasn't a citizen of this country, for one thing", Trump replied.
"If you look at Saudi Arabia, they're an ally, and they're a tremendous purchaser of, not only military equipment, but other things", Trump said Wednesday.
On a day when foreign policy experts worldwide were nearly uniformly accusing Saudi Arabia's government of murdering a prominent dissident, President Donald Trump spoke to the Saudi king, then offered an alternative theory: "Rogue killers" may be to blame.
There was also new political pressure on Trump with nine senators from the opposition Democrats writing to express "significant concerns about conflicts of interest" between Trump and Saudi Arabia concerning deals done through his real estate empire. Saudi Interior Minister Prince Abdulaziz bin Saud bin Naif has condemned the "lies and baseless allegations" against the kingdom, although he praised the joint investigation with Turkey. Saudi Arabia is engaged in a bloody civil war in Yemen that has killed thousands of civilians and exacerbated a starvation that has killed many more.
"They indicated they understood that getting that done in a timely, rapid fashion so they could begin to answer important questions", he told reporters traveling with him.
The Saudis have denied involvement in the disappearance of regime-critic Mr Khashoggi, who went to the consulate to collect a divorce certificate. The Secretary will also discuss the Jamal Khashoggi case and reiterate the U.S. offer to assist Turkey in its investigation, the State Department spokeswoman Heather Nauert said in a statement.