The lawyer investigating alleged links between Donald Trump's election campaign and Russia is examining transactions involving the US President's businesses, it has been reported.

Special counsel Robert Mueller is also expected to look into the activities of Trump associates and family members, despite the President claiming it would be a "violation" if the probe encompassed his personal finances.

The billionaire this week warned the lawyer off investigating his family finances, amid reports his own legal team was digging into Mr Mueller's background looking for potential conflicts of interest.

But Bloomberg reported investigators were already looking into deals with Russians including apartment purchases in his buildings, a controversial New York development project, the multimillion-dollar sale of a Florida home, and the 2013 Miss Universe pageant held in Moscow.

The special counsel is reportedly also interested in transactions with one of the Trump empire's main lenders, Deutsche Bank.

Several of Mr Trump's family members and close advisers have become ensnared in the investigation, including son Donald Trump Jr and son-in-law and White House senior adviser Jared Kushner.

Probing the family's sprawling business ties would bring a probe the President has called a partisan "witch hunt" even closer to the Oval Office.

Donald Trump introduces his eldest son, who is likely to be scrutinised by Mr Mueller's probe, during last year's election campaign (Brian Snyder/Reuters)

​In an interview with the New York Times on Wednesday, Mr Trump insisted he does not "make money from Russia". Asked if Mr Mueller investigating his finances would be "a breach of what his actual charge is," he answered: "I would say yes."

"I think that's a violation," he added.

“The President is making clear that the special counsel should not move outside the scope of the investigation,” said White House spokeswoman Sarah Huckabee Sanders on Thursday.

Mr Trump told the newspaper he was aware that members of Mr Mueller's investigative team had potential conflicts of interest and would disclose the information "at some point".

Mr Trump's lawyers have been delving into the political affiliations and work history of investigators, a source told the Associated Press anonymously.

The President's attorney Jay Sekulow said Mr Trump's lawyers "will consistently evaluate the issue of conflicts and raise them in the appropriate venue".

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Ms Huckabee Sanders said on Thursday that Mr Trump had no intention of firing the special counsel "at this time," but she did not rule out doing so in the future. She also reiterated Mr Trump's concern about the scope of Mueller's investigation, saying it "should stay in the confines of meddling, Russia meddling, and the election and nothing beyond that".

But California Representative Adam Schiff, the top Democrat on the House intelligence committee, said Mr Mueller had the authority to investigate any ties the Trump family has to Russia, "including financial, and anything that arises".

"That is his duty," he added.

William Jeffress, a longtime defence attorney at Baker Botts who represented former President Richard Nixon, said Mr Mueller's inquiry would almost certainly involve examining financial information as he looks for any connections between Trump associates and Russia.

He suggested the President's threats towards the special counsel were not helping his case, adding: "If I were his lawyer, I would be telling him to dial it down."