Donald Trump has been warned he faces the "very real prospect" of jail time as the fallout from the bombshell publication of court filings intensifies.

As the probe into Mr Trump's conduct both before and during office escalates, House Democrats are openly raising the prospect of impeachment or prison time for the US president if it is proved he directed illegal hush-money payments to women.

Jerry Nadler, the incoming chairman of the house judiciary committee, described the details in prosecutors' filings in the case of Mr Trump's former personal lawyer, Michael Cohen, as evidence Mr Trump was "at the centre of a massive fraud".

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"They would be impeachable offences," Mr Nadler said.

"There's a very real prospect that on the day Donald Trump leaves office, the Justice Department may indict him, that he may be the first president in quite some time to face the real prospect of jail time," said Adam Schiff, the incoming chairman of the House intelligence committee.

"The bigger pardon question may come down the road as the next president has to determine whether to pardon Donald Trump."

In the filings, prosecutors in New York for the first time link Mr Trump to a federal crime of illegal payments to buy the silence of two women during the 2016 campaign.

When asked what usually happened after such a filing, James Comey, the sacked former FBI director, told MSNBC: “That person would be in serious jeopardy of being charged.”

Special counsel Robert Mueller's office also laid out previously undisclosed contacts between Trump associates and Russian intermediaries and suggested the Kremlin aimed early on to influence Mr Trump and his Republican campaign by playing to both his political and personal business interests.

Mr Trump has denied wrongdoing and has compared the investigations to a "witch hunt."

Mr Nadler said it was too early to say whether Congress would pursue impeachment proceedings based on the illegal payments alone because lawmakers would need to weigh the gravity of the offence to justify "overturning" the 2016 election.

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Mr Nadler and other lawmakers said on Sunday that they would await additional details from Mr Mueller's investigation into Russian election interference and possible coordination with the Trump campaign to determine the extent of Mr Trump's misconduct.

Regarding the illegal payments, "whether they are important enough to justify an impeachment is a different question, but certainly they'd be impeachable offences because even though they were committed before the president became president, they were committed in the service of fraudulently obtaining the office," Mr Nadler said.

Mr Mueller has not said when he will complete a report of any findings, and it isn't clear that any such report would be made available to Congress. That would be up to the attorney general. Mr Trump on Friday said he would nominate former attorney general William Barr to the post to succeed Jeff Sessions.

Mr Nadler indicated that Democrats, who will control the House in January, will step up their own investigations. He said Congress, the Justice Department and the special counsel needed to dig deeper into the allegations, which included questions about whether Mr Trump lied about his business arrangements with Russians and about possible obstruction of justice.

"The new Congress will not try to shield the president," he said. "We will try to get to the bottom of this, in order to serve the American people and to stop this massive conspiracy — this massive fraud on the American people."

Mr Schiff also stressed a need to wait "until we see the full picture." He has previously indicated his panel would seek to look into the Trump family's business ties with Russia.

"I think we also need to see this as a part of a broader pattern of potential misconduct by the president, and it's that broad pattern, I think, that will lead us to a conclusion about whether it rises to the level to warrant removal from office," Mr Schiff said.

In the legal filings, the Justice Department stopped short of accusing Mr Trump of directly committing a crime. But it said Mr Trump told Mr Cohen to make illegal payments to porn actress Stormy Daniels and former Playboy model Karen McDougal, both of whom claimed to have had affairs with Mr Trump more than a decade ago.

In separate filings, Mr Mueller's team detail 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."'

Mr Cohen said he never followed up on that meeting. Mr Mueller's team also said former campaign chairman Paul Manafort lied to them about his contacts with a Russian associate and Trump administration officials, including in 2018.

Republican senator Marco Rubio called the latest filings "relevant" in judging Mr Trump's fitness for office but said lawmakers need more information to render judgment. He also warned the White House about considering a pardon for Mr Manafort, saying such a step could trigger congressional debate about limiting a president's pardon powers.

Such a move would be "a terrible mistake," Mr Rubio said. "Pardons should be used judiciously. They're used for cases with extraordinary circumstances."

Senator Angus King, a member of the Senate intelligence committee, cautioned against a rush to impeachment, which he said citizens could interpret as "political revenge and a coup against the president",

"The best way to solve a problem like this, to me, is elections," Mr King said. "I'm a conservative when it comes to impeachment. I think it's a last resort and only when the evidence is clear of a really substantial legal violation. We may get there, but we're not there now."

Democratic senator Chris Murphy of Connecticut urged Mr Mueller to "show his cards soon" so that Congress can make a determination early next year on whether to act on impeachment.

"Let's be clear: We have reached a new level in the investigation," Mr Murphy said. "It's important for Congress to get all of the underlying facts and data and evidence that the special counsel has."

Mr Nadler spoke on CNN's State of the Union on Sunday, Mr Rubio was on CNN and ABC's This Week, and Mr Schiff appeared on CBS' Face the Nation. Mr Murphy spoke on ABC, and Mr King was on NBC's Meet the Press.

Additional reporting by AP