However, Trump himself has repeatedly stated that he would like to speak with Mueller about the ongoing investigation, which is examining his campaign's possible collusion with Russia over election interference, and possible obstruction of justice.
"I'm looking forward to it, actually," Trump told reporters at the White House last month, though adding: "subject to my lawyers and all of that."
The president said he would even testify under oath. While Mueller's questioning would not be under oath, it is a crime to lie to federal investigators.
The New York Times cited four people briefed on the matter as saying the president's lawyers are concerned that he could be charged with lying to investigators, as he has previously made false statements and contradicted himself.
The sources said lawyer John Dowd, his deputy Jay Sekulow, longtime personal lawyer Marc Kasowitz and many West Wing advisers want to resist the interview request, claiming Mueller lacks the legal standing on some of the issues he is investigating.
However, a refusal could lead Mueller to issue a subpoena for the president to testify before a grand jury, with a court fight that would be decided by the US Supreme Court.
It could also trigger accusations that Trump is hiding information from the public, a painful distraction for Republicans seeking office in November's mid-term elections.
The Times said Dowd has been discussing the issue of a potential interview with Mueller's office since December, adding that White House lawyer Ty Cobb is one of the few Trump aides arguing for cooperating.
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