The Washington Post, citing unidentified officials, carried the report on Wednesday.
Mr Mueller is leading the Russian probe after being appointed by Deputy Attorney-General Rod Rosenstein in the Department of Justice, which oversees the Federal Bureau of Investigation. Last week, former FBI Director James Comey told Congress he believes he was fired by Mr Trump to undermine the agency's Russia probe.
The obstruction of justice investigation into Mr Trump began days after Mr Comey was fired on May 9, according to people familiar with the matter, The Washington Post said.
Mark Corallo, a spokesman for Mr Trump's legal team, on Wednesday denounced the report, saying: "The FBI leak of information regarding the President is outrageous, inexcusable and illegal."
A spokesman for Mr Mueller's team declined to comment on Wednesday.
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An obstruction of justice finding could form the basis for Mr Trump's impeachment but such a step would face a steep hurdle because it would require approval by the US House of Representatives, which is controlled by Mr Trump's fellow Republicans.
Mr Trump previously claimed he was the victim of a "witch hunt" in May after Mr Mueller, a former FBI director, was appointed to lead an investigation into his campaign's alleged ties to Russia.
"This is the single greatest witch hunt of a politician in American history!" the US president tweeted.
Mr Trump's tweet sparked the derision of social media users.
Seth Moulton, a congressman from Salem in Massachusetts, tweeted, "As the Representative of Salem, MA, I can confirm that this is false," referring to Mr Trump's "witch hunt" whine.
Salem is famously known for farcical trials between February 1692 and May 1693 in which people were accused of witchcraft.
More than 200 were accused and five died in custody. Nineteen were hanged.
