Robert Mueller is believed to want to see documents that relate to 13 “areas of interest”, including those relating to Mr Trump’s sacking of the FBI director James Comey and his son’s meeting with a Russian lawyer, according to US reports.
Mr Mueller is also believed to be interested in any files relating to the sacking of Michael Flynn, the national security adviser, in February.
The retired lieutenant general was fired in February after it was revealed he misled Vice President Mike Pence about December phone calls he had with the Russian ambassador at the time Sergey Kislyak.
Mr Flynn has claimed he had only exchanged holiday greetings with Mr Kislyak when in reality they had discussed economic sanctions against Russia.
Donald Trump branded claims his aides colluded with Moscow as fake news.
Mr Mueller is also said to be looking into the Oval Office meeting in which Mr Trump told Russian officials firing “nut job” FBI chief James Comey in May had relieved “great pressure” on himself.
The special counsel is also believed to have asked for documents relating to the White House response to questions about Donald Trump Jr’s June 2016 Trump Tower meeting with a Russian lawyer.
Mr Trump’s eldest son was led to believe the meeting would yield damaging information about his father’s Democratic presidential opponent Hillary Clinton.
Mr Trump’s lawyer Ty Cobb told Mr Mueller’s office he would turn over several documents this week, according to the New York Times.
Mr Cobb said in a statement: “We can’t comment on any specific requests being made or our conversations with the special counsel.”
Mr Mueller has also reportedly asked for any White House emails or documents relating to Paul Manafort, the former chief of the Trump campaign, the Washington Post claimed, as well as a number of other former campaign officials.
Mr Manafort quit the campaign before the election amid questions about his work for a Ukrainian political party linked to the Russian government.
The US media made claims Mr Manafort was wiretapped by the FBI over concerns about his links with Moscow.
Security agents from the FBI raided Mr Manafort’s home in Washington in July, with agents said to have picked his locks whilst he was in bed.
Mr Mueller’s investigation along with several congressional committees and the Justice Department are all looking at US intelligence findings indicate Russian-backed hackers tried to sway the presidential election held last November in favour of Mr Trump.
There appears to be no indication Mr Mueller is trying to investigate Mr Trump’s private finances or business dealings, areas the president has previously said should be off limits.