US President Donald Trump has issued an ominous warning about the Justice Department and the FBI, promising further firings to get rid of a "lingering stench".

The threat followed reports that deputy attorney general Rod Rosenstein discussed secretly recording him.

Trump, speaking at a rally in Missouri, did not explicitly mention the Rosenstein furore, which was first reported by The New York Times and confirmed by The Associated Press.

But Trump lashed out against what he sees as anti-Trump bias in the Justice Department, touting the firings he has orchestrated, unnerving many in federal law enforcement and sparking fears about the future of the special counsel's Russia probe, which Rosenstein oversees.

"You've seen what happened in the FBI and the Department of Justice. The bad ones, they're all gone. They're all gone," Trump said.

"But there is a lingering stench and we're going to get rid of that, too."

One person present during Rosenstein's remarks said he was being sarcastic.

The Times also said he raised the idea of using the 25th Amendment to remove Trump as unfit for office.

Rosenstein said the story is "inaccurate and factually incorrect".

It was the latest storm to buffet the White House, which this week was battered by a number of potentially damaging stories.

Beyond the speculation about Rosenstein, it was revealed that Trump's former fixer, Michael Cohen, is co-operating with special counsel Robert Mueller's investigation.

The president also backed off his plan to declassify documents related to the ongoing probe, and the fate of his Supreme Court nominee, Judge Brett Kavanaugh, remained uncertain.

Negotiations continued about a possible appearance by Christine Blasey Ford, the woman who accused Judge Kavanaugh of sexual assault at a high school party more than three decades ago.

Trump made the future of Kavanaugh and the federal judiciary a centrepiece of his rally in Springfield, which was designed to support the state's Republican Senate candidate, Missouri attorney general Josh Hawley, in his race against Democratic incumbent Claire McCaskill.