New York (CNN) Federal prosecutors said for the first time Friday that Michael Cohen acted at the direction of Donald Trump when the former fixer committed two election-related crimes during the 2016 presidential campaign, as special counsel Robert Mueller outlined a previously undisclosed set of overtures and contacts between the Trump campaign and Russian nationals.

Those revelations came in a set of court filings in which federal prosecutors in New York said Cohen should receive a "substantial" prison sentence of roughly four years for tax fraud and campaign-finance crimes, and as Mueller's office accused the President's former attorney of lying to them about his contacts with Russia.

Mueller's disclosures also exposed deeper entanglements than previously known between Trump, his campaign apparatus and the Russian government, including that a Russian national who claimed to be well-connected in Moscow spoke with Cohen in 2015 and offered "political synergy" with the Trump campaign.

In August, Cohen pleaded guilty to eight federal crimes after being charged by Manhattan federal prosecutors. Those included tax fraud, making false statements to a bank and campaign-finance violations tied to his work for Trump, including payments Cohen made or helped orchestrate that were designed to silence women who claimed affairs with the then-presidential candidate. Trump has denied those claims.

On Friday, prosecutors wrote, "In particular, and as Cohen himself has now admitted, with respect to both payments, he acted in coordination with and at the direction of Individual-1." Individual-1 is the term prosecutors have been using to refer to the President.

The pair of memos from two sets of prosecutors reflected competing views of Cohen's utility to the federal investigations ahead of his scheduled sentencing on Wednesday.

In their filing, prosecutors from the Manhattan US attorney's office knocked Cohen's "rose-colored view of the seriousness of his crimes," noting his years-long willingness to break the law. "He was motivated to do so by personal greed, and repeatedly used his power and influence for deceptive ends," they wrote in a filing that also sought to puncture Cohen's public pronouncements of assistance in their investigations.

Mueller's team acknowledged that Cohen provided "useful information concerning certain discrete Russia-related matters core to its investigation," but also disclosed that he had initially lied to them about a proposed Trump Tower project in Moscow.

Contact with Russians

In addition to the August case, Cohen was subsequently charged last week by the special counsel's office with one count of lying to Congress about the Moscow project. He pleaded guilty, disclosing that talks about the effort in Moscow had extended through June 2016, after Trump had become the presumptive Republican nominee for president, and that both Trump and his family members had been briefed on the discussions. Cohen also acknowledged pursuing plans to send Trump and himself to Russia in service of the project and discussing the proposed development directly with a representative of the Kremlin.

The special counsel's Friday memo on the matter suggested that Cohen's -- and, by extension, Trump's -- intersection with Russians eager to aid the presidential campaign was more substantial than previously known, with the Russians anxious to put Trump face-to-face with their president.

In one instance, Cohen spoke in November 2015 to a Russian national who offered "synergy on a government level" to the Trump presidential campaign and "repeatedly proposed" a meeting between Trump and Russian President Vladimir Putin, according to court papers.

The interest appears to have been reciprocal. Cohen told the special counsel that he had consulted with Trump about his interest in contacting the Russian government before suggesting in a radio interview in September 2015 that Trump meet with Putin during the Russian President's visit to New York City that fall.

Cohen had previously claimed his comments on air were spontaneous, the court papers said, but he admitted to prosecutors that they came about after his discussion with Trump.

That meeting never happened, nor did the one with the Russian national, although Cohen told the special counsel that was due to the fact that he was working on the Moscow project "with a different individual who Cohen understood to have his own connections to the Russian government."

Mueller's sentencing memo also provided the first explanation of the Trump Tower Moscow project's relevance to Russia's interference during the 2016 campaign.

According to the special counsel, Cohen's false statements to investigators about the Trump Tower Moscow project "obscured the fact that the Moscow Project was a lucrative business opportunity that sought, and likely required, the assistance of the Russian government."

That Cohen continued to work on the Trump Tower Moscow project -- and discuss it with Trump -- was material to both the ongoing congressional and special counsel investigations, prosecutors said, noting in particular that "it occurred at a time of sustained efforts by the Russian government to interfere with the U.S. presidential election."

A tightening net

The filings on Cohen, and a submission from Mueller about Trump's former campaign manager Paul Manafort, appear to further tighten the net that Mueller is gathering around the President and his inner circle.

They come at a time when Trump appears to be increasingly worried and furious about the investigation, following a searing tweet storm aimed at the special counsel on Friday morning. White House press secretary Sarah Sanders downplayed the significance of prosecutors' findings, saying late Friday that "the government's filings in Mr. Cohen's case tell us nothing of value that wasn't already known. Mr. Cohen has repeatedly lied and as the prosecution has pointed out to the court, Mr. Cohen is no hero."

The memos also appear to take aim at several issues that have and could continue to ensnare others in the President's orbit, including the act of lying not just to federal agents but also to the public and the media. Cohen's effort to lie about the Moscow project, they said, constituted an attempt to alter the course of the investigation.

"The defendant amplified his false statements by releasing and repeating his lies to the public, including to other potential witnesses," prosecutors wrote. "By publicly presenting this false narrative, the defendant deliberately shifted the timeline of what occurred in the hopes of limiting the investigation into possible Russian interference in the 2016 US presidential election -- an issue of heightened national interest."

Prosecutors unhappy with Cohen's cooperation

Despite Cohen's disclosures to the special counsel, however, Manhattan federal prosecutors sought repeatedly to emphasize the limited nature of his assistance in other investigations.

New York federal prosecutors have continued to investigate the Trump Organization based on information in the charges against Cohen, CNN has reported, and Cohen met with prosecutors to discuss "the participation of others in the campaign finance crimes," they said in their filing. But, they noted, he "declined to meet with the office about other areas of investigative interest."

Prosecutors also said the information Cohen has provided to the New York State Attorney General's Office, which has a civil lawsuit against the Donald J. Trump Foundation, "warrants little to no consideration as a mitigating factor."

And while Cohen has met with the New York State Department of Taxation and Finance, which has also been examining the Trump Organization, CNN has reported, his assistance "appears to consist solely of providing that entity information that they would otherwise have obtained via subpoena."

The Cohen cases are being considered together for sentencing purposes. In the New York federal prosecutors' case, the sentencing guidelines call for a term of 51 to 63 months in prison, and prosecutors on Friday asked the court to "impose a substantial term of imprisonment, one that reflects a modest downward variance from the applicable guidelines range."

US District Judge William Pauley III can deviate from the guidelines when he determines Cohen's sentence.

Cohen's attorneys have asked the court to give him no prison time, but prosecutors on Friday voiced their objection to that proposal, noting that the crimes Cohen committed had far-reaching consequences that extended all the way to the White House.

"While many Americans who desired a particular outcome to the election knocked on doors, toiled at phone banks, or found any number of other legal ways to make their voices heard, Cohen sought to influence the election from the shadows," they wrote.

"He did so by orchestrating secret and illegal payments to silence two women who otherwise would have made public their alleged extramarital affairs with Individual-1," they said. "In the process, Cohen deceived the voting public by hiding alleged facts that he believed would have had a substantial effect on the election."