He had provided information about contacts with Russian interests during the campaign, attempts by Russians to reach the campaign and about contacts with "persons connected to the White House" in 2017-2018, the filing added.
"In total, the prosecutors seem to be saying the president was more aware than he has claimed to be", former federal prosecutor Michael Zeldin said.
Cohen is scheduled to be sentenced next Wednesday by U.S. District Judge William Pauley in Manhattan on all of the charges to which he pleaded guilty.
Prosecutors in NY, where Cohen pleaded guilty in August to campaign finance crimes in connection with those payments, said the lawyer "acted in coordination and at the direction" of Trump.
In the 40-page document Khuzami drew a direct link between Cohen's illegal behavior and Trump.
The campaign finance violations to which Cohen plead guilty - unrelated to the Russian Federation investigation - concerned hush payments he made on Trump's behalf to alleged former lovers of the president, including porn star Stormy Daniels.
"When you look at what the prosecutors did in the Southern District of NY they allege that Donald Trump directed two campaign finance violations".More news: France tops OECD list of most taxed countries
Cohen told prosecutors he never followed up on the Putin invitation, though the offer bore echoes of a March 2016 proposal presented by Trump campaign aide George Papadopoulos, who broached to other advisers the idea of a Putin encounter.
The White House similarly dismissed that filing, arguing it "says absolutely nothing about the President".
Trump's spokeswoman Sarah Sanders dismissed the latest filings in Cohen's case, saying they "tell us nothing of value that wasn't already known". He has said he did so to stay in sync with Trump's political messaging, and that he consulted with the White House while preparing to testify to Congress. Prosecutors in NY, where Cohen pleaded guilty in August in connection with those payments, said the lawyer "acted in coordination and at the direction" of Trump. Cohen also said he discussed the project with Trump during the presidential campaign, undercutting the then-Republican presidential candidate's statements that he didn't have any deals in Russian Federation.
Mueller said that discussions about the potential Moscow development were relevant to the investigation because they occurred "at a time of sustained efforts by the Russian government to interfere with the USA presidential election".
The commander-in-chief vowed his lawyers would produce a "major Counter Report" to rebut Mueller's findings, as and when he delivers them.
Mueller also detailed alleged lies told by Manafort during interviews with prosecutors and the Federal Bureau of Investigation.More news: Trump ghostwriter ridicules belief John Kelly’s replacement will reel president in
The heavily redacted report came more than a week after prosecutors accused Manafort of "committing federal crimes by lying to the Federal Bureau of Investigation and the special counsel's office on a variety of subject matters", violating his plea agreement.
In an additional filing Friday evening, prosecutors said former Trump campaign chairman Paul Manafort lied to them about his contacts with a Russian associate and Trump administration officials.
"Mr Cohen has repeatedly lied and as the prosecution has pointed out to the court, Mr Cohen is no hero", she said.
Cohen continued to provide "relevant and truthful information" to assist the probe, holding seven sessions with investigators, "many of them lengthy, and continues to make himself available to investigators", it said.
Prosecutors said the court's Probation Department estimated that federal sentencing guidelines call for Cohen to serve at least four years in prison.
Mueller, for his part, praised Cohen for voluntarily providing information about his own and others' conduct on "core topics under investigation" and described the information as "credible and consistent with other evidence" they had obtained.More news: PSG set to sign Ajax star De Jong in record deal
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