The filings, in cases involving Mr. Cohen and former campaign chairman Paul Manafort, capped a dramatic week of revelations in Mr. Mueller’s probe into possible coordination between the Trump campaign and the Kremlin.

President Donald Trump’s former lawyer, Michael Cohen, was in touch as far back as 2015 with a Russian who offered “political synergy” with the Trump election campaign and proposed a meeting between the candidate and Russian President Vladimir Putin, the federal special counsel said.

Court filings from prosecutors in New York and special counsel Robert Mueller’s office on Friday laid out previously undisclosed contacts between Mr. Trump associates and Russian intermediaries and suggested the Kremlin aimed early on to influence Mr. Trump and his campaign by playing to both his political aspirations and his personal business interests.

The filings, in cases involving Mr. Cohen and former campaign chairman Paul Manafort, capped a dramatic week of revelations in Mr. Mueller’s probe into possible coordination between the Trump campaign and the Kremlin. They bring the legal peril from multiple investigations closer than ever to Mr. Trump, tying him to an illegal hush money payment scheme and contradicting his claims that he had nothing to do with Russia.

They make clear how witnesses previously close to Mr. Trump — Mr. Cohen once declared he’d “take a bullet” for the President — have since provided damaging information about him in efforts to come clean to the government and in some cases get lighter prison sentences.

One defendant, former National Security Adviser Michael Flynn, provided so much information to prosecutors that Mr. Mueller this week said he shouldn’t serve any prison time.

In hours of interviews with prosecutors, witnesses have offered up information about pivotal episodes under examination, including possible collusion with Russia and payments during the campaign to silence a porn star and Playboy model who said they had sex with Mr. Trump a decade earlier.

‘Political synergy’

In one of the filings, Mr. Mueller details how Mr. Cohen spoke to a Russian who “claimed to be a ‘trusted person’ in the Russian Federation who could offer the campaign ‘political synergy’ and ‘synergy on a government level.’”

The person repeatedly dangled a meeting between Mr. Trump and Mr. Putin, saying such a meeting could have a “phenomenal” impact “not only in political but in a business dimension as well.”

That was a reference to a proposed Moscow real estate deal that prosecutors say could have netted Mr. Trump’s business hundreds of millions of dollars. Mr. Cohen admitted last week to lying to the Congress by saying discussions about a Trump Tower in Moscow ended in January 2016 when in fact they stretched into that June, well into the U.S. campaign.

Mr. 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.

Prosecutors said probation officials recommended a sentence for Mr. Cohen of three-and-a-half years in prison. His lawyers want the 52-year-old attorney to avoid prison time altogether.

In an additional filing on Friday evening, prosecutors said Mr. Manafort lied about his contacts with a Russian associate and Trump administration officials, including in 2018.

The court papers say Mr. Manafort initially told prosecutors he didn’t have contact with any people while they were in the Trump administration. But prosecutors say they recovered “electronic documents” showing contacts with multiple administration officials not identified in the filings.

Mr. Manafort, who has pleaded guilty to several counts, violated his plea agreement by telling “multiple discernible lies” to prosecutors, they said.

Mr. Manafort resigned from his job on the Trump campaign as questions swirled about his lobbying work for a pro-Russia political party in Ukraine.

'Manifold crimes'

Prosecutors in Mr. Cohen’s case said that even though he cooperated in their investigation into potential campaign finance violations, he nonetheless deserved prison time. Though he has portrayed himself as cooperative, “his description of those efforts is overstated in some respects and incomplete in others,” prosecutors said.

“After cheating the IRS for years, lying to banks and to Congress, and seeking to criminally influence the Presidential election, Mr. Cohen’s decision to plead guilty — rather than seek a pardon for his manifold crimes — does not make him a hero,” they wrote.

Mr. Cohen, dubbed Mr. Trump’s “legal fixer” in the past, also described his work in conjunction with Mr. Trump in orchestrating hush money payments to two women adult actress Stormy Daniels and Playboy model Karen McDougal who said they had sex with Mr. Trump.

Prosecutors in New York, where Mr. 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 Mr. Trump. Though Mr. Cohen had previously implicated Mr. Trump in the payments, the prosecutors now are linking Mr. Trump to the scheme and backing up Mr. Cohen’s allegations.

Federal law requires that any payments made “for the purposes of influencing” an election must be reported in campaign finance disclosures. The court filing on Friday makes clear that the payments were made to benefit Mr. Trump politically.

Mr. Trump tried to brush off Friday’s revelations, claiming wrongly on Twitter that the news “Totally clears the President. Thank you!”

A court filing also reveals that Mr. Cohen told prosecutors he and Mr. Trump discussed a potential meeting with Mr. Putin on the sidelines of the U.N. General Assembly in 2015, shortly after Mr. Trump announced his candidacy for President. In a footnote Mr. Mueller’s team writes that Mr. Cohen conferred with Mr. Trump “about contacting the Russia government before reaching out to gauge Russia’s interest in such a meeting.” It never took place.